Using the numbers in the contingency table calculate the percentage of antibiotic users who tested positive for candiduria.
6. Using the numbers in the contingency table calculate the percentage of non-antibiotic users who tested positive for candiduria.

Exercise 29
Calculating Simple Linear Regression
Simple linear regression is a procedure that provides an estimate of the value of a dependent variable (outcome) based on the value of an independent variable (predictor). Knowing that estimate with some degree of accuracy we can use regression analysis to predict the value of one variable if we know the value of the other variable (Cohen & Cohen 1983). The regression equation is a mathematical expression of the influence that a predictor has on a dependent variable based on some theoretical framework. For example in Exercise 14 Figure 14-1 illustrates the linear relationship between gestational age and birth weight. As shown in the scatterplot there is a strong positive relationship between the two variables. Advanced gestational ages predict higher birth weights.
A regression equation can be generated with a data set containing subjects’ x and y values. Once this equation is generated it can be used to predict future subjects’ y values given only their x values. In simple or bivariate regression predictions are made in cases with two variables. The score on variable y (dependent variable or outcome) is predicted from the same subject’s known score on variable x (independent variable or predictor).
Research Designs Appropriate for Simple Linear Regression
Research designs that may utilize simple linear regression include any associational design (Gliner etal. 2009). The variables involved in the design are attributional meaning the variables are characteristics of the participant such as health status blood pressure gender diagnosis or ethnicity. Regardless of the nature of variables the dependent variable submitted to simple linear regression must be measured as continuous at the interval or ratio level.
Statistical Formula and Assumptions
Use of simple linear regression involves the following assumptions (Zar 2010):
1. Normal distribution of the dependent (y) variable
2. Linear relationship between x and y
3. Independent observations
4. No (or little) multicollinearity
5. Homoscedasticity
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Data that are homoscedastic are evenly dispersed both above and below the regression line which indicates a linear relationship on a scatterplot. Homoscedasticity reflects equal variance of both variables. In other words for every value of x the distribution of y values should have equal variability. If the data for the predictor and dependent variable are not homoscedastic inferences made during significance testing could be invalid (Cohen & Cohen 1983; Zar 2010). Visual examples of homoscedasticity and heteroscedasticity are presented in Exercise 30.
In simple linear regression the dependent variable is continuous and the predictor can be any scale of measurement; however if the predictor is nominal it must be correctly coded. Once the data are ready the parameters a and b are computed to obtain a regression equation. To understand the mathematical process recall the algebraic equation for a straight line:
y=bx+a

where
y=thedependentvariable(outcome)

x=theindependentvariable(predictor)

b=theslopeoftheline

a=y-intercept(thepointwheretheregressionlineintersectsthey-axis)

No single regression line can be used to predict with complete accuracy every y value from every x value. In fact you could draw an infinite number of lines through the scattered paired values (Zar 2010). However the purpose of the regression equation is to develop the line to allow the highest degree of prediction possiblethe line of best fit. The procedure for developing the line of best fit is the method of least squares. The formulas for the beta () and slope () of the regression equation are computed as follows. Note that once the is calculated that value is inserted into the formula for .
=nxyxynx2(x)2

=ybxn

Hand Calculations
This example uses data collected from a study of students enrolled in a registered nurse to bachelor of science in nursing (RN to BSN) program (Mancini Ashwill & Cipher 2014). The predictor in this example is number of academic degrees obtained by the student prior to enrollment and the dependent variable was number of months it took for the student to complete the RN to BSN program. The null hypothesis is Number of degrees does not predict the number of months until completion of an RN to BSN program.
The data are presented in Table 29-1. A simulated subset of 20 students was selected for this example so that the computations would be small and manageable. In actuality studies involving linear regression need to be adequately powered (Aberson 2010; Cohen 1988). Observe that the data in Table 29-1 are arranged in columns that correspond to 321the elements of the formula. The summed values in the last row of Table 29-1 are inserted into the appropriate place in the formula for b.
TABLE 29-1
ENROLLMENT GPA AND MONTHS TO COMPLETION IN AN RN TO BSN PROGRAM

The computations for the b and are as follows:
Step 1: Calculate b.From the values in Table 29-1 we know that n = 20 x = 20 y = 267 x2 = 30 and xy = 238. These values are inserted into the formula for b as follows:
b=20(238)(20)(267)20(30)202

b=2.9

Step 2: Calculate .From Step 1 we now know that b = 2.9 and we plug this value into the formula for .
=267(2.9)(20)20

=16.25

Step 3: Write the new regression equation:
y=2.9x+16.25

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Step 4: Calculate R.The multiple R is defined as the correlation between the actual y values and the predicted y values using the new regression equation. The predicted y value using the new equation is represented by the symbol to differentiate from y which represents the actual y values in the data set. We can use our new regression equation from Step 3 to compute predicted program completion time in months for each student using their number of academic degrees prior to enrollment in the RN to BSN Program. For example Student #1 had earned 1 academic degree prior to enrollment and the predicted months to completion for Student 1 is calculated as:
y=2.9(1)+16.25

y=13.35

Thus the predicted is 13.35 months. This procedure would be continued for the rest of the students and the Pearson correlation between the actual months to completion (y) and the predicted months to completion () would yield the multiple R value. In this example the R = 0.638. The higher the R the more likely that the new regression equation accurately predicts y because the higher the correlation the closer the actual y values are to the predicted values. Figure 29-1 displays the regression line where the x axis represents possible numbers of degrees and the y axis represents the predicted months to program completion ( values).

FIGURE 29-1 REGRESSION LINE REPRESENTED BY NEW REGRESSION EQUATION.
Step 5: Determine whether the predictor significantly predicts y.
t=Rn21R2

To know whether the predictor significantly predicts y the beta must be tested against zero. In simple regression this is most easily accomplished by using the R value from Step 4:
t=.63820021.407

t=3.52

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The t value is then compared to the t probability distribution table (see Appendix A). The df for this t statistic is n 2. The critical t value at alpha () = 0.05 df = 18 is 2.10 for a two-tailed test. Our obtained t was 3.52 which exceeds the critical value in the table thereby indicating a significant association between the predictor (x) and outcome (y).
Step 6: Calculate R2.After establishing the statistical significance of the R value it must subsequently be examined for clinical importance. This is accomplished by obtaining the coefficient of determination for regressionwhich simply involves squaring the R value. The R2 represents the percentage of variance explained in y by the predictor. Cohen describes R2 values of 0.02 as small 0.15 as moderate and 0.26 or higher as large effect sizes (Cohen 1988). In our example the R was 0.638 and therefore the R2 was 0.407. Multiplying 0.407 100% indicates that 40.7% of the variance in months to program completion can be explained by knowing the student’s number of earned academic degrees at admission (Cohen & Cohen 1983).The R2 can be very helpful in testing more than one predictor in a regression model. Unlike R the R2 for one regression model can be compared with another regression model that contains additional predictors (Cohen & Cohen 1983). The R2 is discussed further in Exercise 30.The standardized beta () is another statistic that represents the magnitude of the association between x and y. has limits just like a Pearson r meaning that the standardized cannot be lower than 1.00 or higher than 1.00. This value can be calculated by hand but is best computed with statistical software. The standardized beta () is calculated by converting the x and y values to z scores and then correlating the x and y value using the Pearson r formula. The standardized beta () is often reported in literature instead of the unstandardized b because b does not have lower or upper limits and therefore the magnitude of b cannot be judged. on the other hand is interpreted as a Pearson r and the descriptions of the magnitude of can be applied as recommended by Cohen (1988). In this example the standardized beta () is 0.638. Thus the magnitude of the association between x and y in this example is considered a large predictive association (Cohen 1988).
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SPSS Computations
This is how our data set looks in SPSS.

Step 1: From the Analyze menu choose Regression and Linear.
Step 2: Move the predictor Number of Degrees to the space labeled Independent(s). Move the dependent variable Number of Months to Completion to the space labeled Dependent. Click OK.

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Interpretation of SPSS Output
The following tables are generated from SPSS. The first table contains the multiple R and the R2 values. The multiple R is 0.638 indicating that the correlation between the actual y values and the predicted y values using the new regression equation is 0.638. The R2 is 0.407 indicating that 40.7% of the variance in months to program completion can be explained by knowing the student’s number of earned academic degrees at enrollment.
Regression

The second table contains the ANOVA table. As presented in Exercises 18 and 33 the ANOVA is usually performed to test for differences between group means. However ANOVA can also be performed for regression where the null hypothesis is that knowing the value of x explains no information about y. This table indicates that knowing the value of x explains a significant amount of variance in y. The contents of the ANOVA table are rarely reported in published manuscripts because the significance of each predictor is presented in the last SPSS table titled Coefficients (see below).

The third table contains the b and a values standardized beta () t and exact p value. The a is listed in the first row next to the label Constant. The is listed in the second row next to the name of the predictor. The remaining information that is important to extract when interpreting regression results can be found in the second row. The standardized beta () is 0.638. This value has limits just like a Pearson r meaning that the standardized cannot be lower than 1.00 or higher than 1.00. The t value is 3.516 and the exact p value is 0.002.

326
Final Interpretation in American Psychological Association (APA) Format
The following interpretation is written as it might appear in a research article formatted according to APA guidelines (APA 2010). Simple linear regression was performed with number of earned academic degrees as the predictor and months to program completion as the dependent variable. The student’s number of degrees significantly predicted months to completion among students in an RN to BSN program = 0.638 p = 0.002 and R2 = 40.7%. Higher numbers of earned academic degrees significantly predicted shorter program completion time.
327
Study Questions
1. If you have access to SPSS compute the Shapiro-Wilk test of normality for months to completion (as demonstrated in Exercise 26). If you do not have access to SPSS plot the frequency distributions by hand. What do the results indicate?
2. State the null hypothesis for the example where number of degrees was used to predict time to BSN program completion.
3. In the formula y = bx + a what does b represent?
4. In the formula y = bx + a what does a represent?
5. Using the new regression equation = 2.9x + 16.25 compute the predicted months to program completion if a student’s number of earned degrees is 0. Show your calculations.
6. Using the new regression equation = 2.9x + 16.25 compute the predicted months to program completion if a student’s number of earned degrees is 2. Show your calculations.
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7. What was the correlation between the actual y values and the predicted y values using the new regression equation in the example?
8. What was the exact likelihood of obtaining a t value at least as extreme as or as close to the one that was actually observed assuming that the null hypothesis is true?
9. How much variance in months to completion is explained by knowing the student’s number of earned degrees?
10. How would you characterize the magnitude of the R2 in the example? Provide a rationale for your answer.
329
Answers to Study Questions
1. The Shapiro-Wilk p value for months to RN to BSN program completion was 0.16 indicating that the frequency distribution did not significantly deviate from normality. Moreover visual inspection of the frequency distribution indicates that months to completion is approximately normally distributed. See SPSS output below for the histograms of the distribution:

2. The null hypothesis is: The number of earned academic degrees does not predict the number of months until completion of an RN to BSN program.
3. In the formula y = bx + a b represents the slope of the regression line.
4. In the formula y = bx + a a represents the y-intercept or the point at which the regression line intersects the y-axis.
5. The predicted months to program completion if a student’s number of academic degrees is 0 is calculated as: = 2.9(0) + 16.25 = 16.25 months.
6. The predicted months to program completion if a student’s number of academic degrees is 2 is calculated as: = 2.9(2) + 16.25 = 10.45 months.
7. The correlation between the actual y values and the predicted y values using the new regression equation in the example also known as the multiple R is 0.638.
8. The exact likelihood of obtaining a t value at least as extreme as or as close to the one that was actually observed assuming that the null hypothesis is true was 0.2%. This value was obtained by looking at the SPSS output table titled Coefficients in the last value of the column labeled Sig.
9. 40.7% of the variance in months to completion is explained by knowing the student’s number of earned academic degrees at enrollment.
10. The magnitude of the R2 in this example 0.407 would be considered a large effect according to the effect size tables in Exercises 24 and 25.
330
Data for Additional Computational Practice for the Questions to be Graded
Using the example from Mancini and colleagues (2014) students enrolled in an RN to BSN program were assessed for demographics at enrollment. The predictor in this example is age at program enrollment and the dependent variable was number of months it took for the student to complete the RN to BSN program. The null hypothesis is: Student age at enrollment does not predict the number of months until completion of an RN to BSN program. The data are presented in Table 29-2. A simulated subset of 20 students was randomly selected for this example so that the computations would be small and manageable.
TABLE 29-2
AGE AT ENROLLMENT AND MONTHS TO COMPLETION IN AN RN TO BSN PROGRAM

331
EXERCISE 29Questions to Be Graded
Name: _______________________________________________________ Class: _____________________
Date: ___________________________________________________________________________________
Follow your instructor’s directions to submit your answers to the following questions for grading. Your instructor may ask you to write your answers below and submit them as a hard copy for grading. Alternatively your instructor may ask you to use the space below for notes and submit your answers online at http://evolve.elsevier.com/Grove/Statistics/ under Questions to Be Graded.
1. If you have access to SPSS compute the Shapiro-Wilk test of normality for the variable age (as demonstrated in Exercise 26). If you do not have access to SPSS plot the frequency distributions by hand. What do the results indicate?
2. State the null hypothesis where age at enrollment is used to predict the time for completion of an RN to BSN program.
3. What is b as computed by hand (or using SPSS)?
4. What is a as computed by hand (or using SPSS)?
332
5. Write the new regression equation.
6. How would you characterize the magnitude of the obtained R2 value? Provide a rationale for your answer.
7. How much variance in months to RN to BSN program completion is explained by knowing the student’s enrollment age?
8. What was the correlation between the actual y values and the predicted y values using the new regression equation in the example?
9. Write your interpretation of the results as you would in an APA-formatted journal.
10. Given the results of your analyses would you use the calculated regression equation to predict future students’ program completion time by using enrollment age as x? Provide a rationale for your answer
(Grove 319-332)
Grove Susan K. Daisha Cipher. Statistics for Nursing Research: A Workbook for Evidence-Based Practice 2nd Edition. Saunders 022016. VitalBook file.
The citation provided is a guideline. Please check each citation for accuracy before use.
Exercise 35
Calculating Pearson Chi-Square
The Pearson chi-square test (2) compares differences between groups on variables measured at the nominal level. The 2 compares the frequencies that are observed with the frequencies that are expected. When a study requires that researchers compare proportions (percentages) in one category versus another category the 2 is a statistic that will reveal if the difference in proportion is statistically improbable.
A one-way 2 is a statistic that compares different levels of one variable only. For example a researcher may collect information on gender and compare the proportions of males to females. If the one-way 2 is statistically significant it would indicate that proportions of one gender are significantly higher than proportions of the other gender than what would be expected by chance (Daniel 2000). If more than two groups are being examined the 2 does not determine where the differences lie; it only determines that a significant difference exists. Further testing on pairs of groups with the 2 would then be warranted to identify the significant differences.
A two-way 2 is a statistic that tests whether proportions in levels of one nominal variable are significantly different from proportions of the second nominal variable. For example the presence of advanced colon polyps was studied in three groups of patients: those having a normal body mass index (BMI) those who were overweight and those who were obese (Siddiqui Mahgoub Pandove Cipher & Spechler 2009). The research question tested was: Is there a difference between the three groups (normal weight overweight and obese) on the presence of advanced colon polyps? The results of the 2 test indicated that a larger proportion of obese patients fell into the category of having advanced colon polyps compared to normal weight and overweight patients suggesting that obesity may be a risk factor for developing advanced colon polyps. Further examples of two-way 2 tests are reviewed in Exercise 19.
Research Designs Appropriate for the Pearson 2
Research designs that may utilize the Pearson 2 include the randomized experimental quasi-experimental and comparative designs (Gliner Morgan & Leech 2009). The variables may be active attributional or a combination of both. An active variable refers to an intervention treatment or program. An attributional variable refers to a characteristic of the participant such as gender diagnosis or ethnicity. Regardless of the whether the variables are active or attributional all variables submitted to 2 calculations must be measured at the nominal level.
410
Statistical Formula and Assumptions
Use of the Pearson 2 involves the following assumptions (Daniel 2000):
1. Only one datum entry is made for each subject in the sample. Therefore if repeated measures from the same subject are being used for analysis such as pretests and posttests 2 is not an appropriate test.
2. The variables must be categorical (nominal) either inherently or transformed to categorical from quantitative values.
3. For each variable the categories are mutually exclusive and exhaustive. No cells may have an expected frequency of zero. In the actual data the observed cell frequency may be zero. However the Pearson 2 test is sensitive to small sample sizes and other tests such as the Fisher’s exact test are more appropriate when testing very small samples (Daniel 2000; Yates 1934).
The test is distribution-free or nonparametric which means that no assumption has been made for a normal distribution of values in the population from which the sample was taken (Daniel 2000).
The formula for a two-way 2 is:
2=n[(A)(D)(B)(C)]2(A+B)(C+D)(A+C)(B+D)

The contingency table is labeled as follows. A contingency table is a table that displays the relationship between two or more categorical variables (Daniel 2000):
With any 2 analysis the degrees of freedom (df) must be calculated to determine the significance of the value of the statistic. The following formula is used for this calculation:
df=(R1)(C1)

where
R=Numberofrows

C=Numberofcolumns

Hand Calculations
A retrospective comparative study examined whether longer antibiotic treatment courses were associated with increased antimicrobial resistance in patients with spinal cord injury (Lee etal. 2014). Using urine cultures from a sample of spinal cordinjured veterans two groups were created: those with evidence of antibiotic resistance and those with no evidence of antibiotic resistance. Each veteran was also divided into two groups based on having had a history of recent (in the past 6 months) antibiotic use for more than 2 weeks or no history of recent antibiotic use.
411
The data are presented in Table 35-1. The null hypothesis is: There is no difference between antibiotic users and non-users on the presence of antibiotic resistance.
TABLE 35-1
ANTIBIOTIC RESISTANCE BY ANTIBIOTIC USE
The computations for the Pearson 2 test are as follows:
Step 1: Create a contingency table of the two nominal variables:

Step 2: Fit the cells into the formula:
2=n[(A)(D)(B)(C)]2(A+B)(C+D)(A+C)(B+D)

2=42[(8)(21)(7)(6)]2(8+7)(6+21)(8+6)(7+21)

2=666792158760

2=4.20

Step 3: Compute the degrees of freedom:
df=(21)(21)=1

Step 4: Locate the critical 2 value in the 2 distribution table (Appendix D) and compare it to the obtained 2 value.
The obtained 2 value is compared with the tabled 2 values in Appendix D. The table includes the critical values of 2 for specific degrees of freedom at selected levels of significance. If the value of the statistic is equal to or greater than the value identified in the 2 table the difference between the two variables is statistically significant. The critical 2 for df = 1 is 3.84 and our obtained 2 is 4.20 thereby exceeding the critical value and indicating a significant difference between antibiotic users and non-users on the presence of antibiotic resistance.
Furthermore we can compute the rates of antibiotic resistance among antibiotic users and non-users by using the numbers in the contingency table from Step 1. The antibiotic resistance rate among the antibiotic users can be calculated as 8 14 = 0.571 100% = 57.1%. The antibiotic resistance rate among the non-antibiotic users can be calculated as 7 28 = 0.25 100% = 25%.
412
SPSS Computations
The following screenshot is a replica of what your SPSS window will look like. The data for subjects 24 through 42 are viewable by scrolling down in the SPSS screen.

413
Step 1: From the Analyze menu choose Descriptive Statistics and Crosstabs. Move the two variables to the right where either variable can be in the Row or Column space.

Step 2: Click Statistics and check the box next to Chi-square. Click Continue and OK.

414
Interpretation of SPSS Output
The following tables are generated from SPSS. The first table contains the contingency table similar to Table 35-1 above. The second table contains the 2 results.
Crosstabs

The last table contains the 2 value in addition to other statistics that test associations between nominal variables. The Pearson 2 test is located in the first row of the table which contains the 2 value df and p value.
Final Interpretation in American Psychological Association (APA) Format
The following interpretation is written as it might appear in a research article formatted according to APA guidelines (APA 2010). A Pearson 2 analysis indicated that antibiotic users had significantly higher rates of antibiotic resistance than those who did not use antibiotics 2(1) = 4.20 p = 0.04 (57.1% versus 25% respectively). This finding suggests that extended antibiotic use may be a risk factor for developing resistance and further research is needed to investigate resistance as a direct effect of antibiotics.
415
Study Questions
1. Do the example data meet the assumptions for the Pearson 2 test? Provide a rationale for your answer.
2. What is the null hypothesis in the example?
3. What was the exact likelihood of obtaining a 2 value at least as extreme or as close to the one that was actually observed assuming that the null hypothesis is true?
4. Using the numbers in the contingency table calculate the percentage of antibiotic users who were resistant.
5. Using the numbers in the contingency table calculate the percentage of non-antibiotic users who were resistant.
6. Using the numbers in the contingency table calculate the percentage of resistant veterans who used antibiotics for more than 2 weeks.
416
7. Using the numbers in the contingency table calculate the percentage of resistant veterans who had no history of antibiotic use.
8. What kind of design was used in the example?
9. What result would have been obtained if the variables in the SPSS Crosstabs window had been switched with Antibiotic Use being placed in the Row and Resistance being placed in the Column?
10. Was the sample size adequate to detect differences between the two groups in this example? Provide a rationale for your answer.
417
Answers to Study Questions
1. Yes the data meet the assumptions of the Pearson 2:
a. Only one datum per participant was entered into the contingency table and no participant was counted twice.
b. Both antibiotic use and resistance are categorical (nominal-level data).
c. For each variable the categories are mutually exclusive and exhaustive. It was not possible for a participant to belong to both groups and the two categories (recent antibiotic user and non-user) included all study participants.
2. The null hypothesis is: There is no difference between antibiotic users and non-users on the presence of antibiotic resistance.
3. The exact likelihood of obtaining a 2 value at least as extreme as or as close to the one that was actually observed assuming that the null hypothesis is true was 4.0%.
4. The percentage of antibiotic users who were resistant is calculated as 8 14 = 0.5714 100% = 57.14% = 57.1%.
5. The percentage of non-antibiotic users who were resistant is calculated as 7 28 = 0.25 100% = 25%.
6. The percentage of antibiotic-resistant veterans who used antibiotics for more than 2 weeks is calculated as 8 15 = 0.533 100% = 53.3%.
7. The percentage of resistant veterans who had no history of antibiotic use is calculated as 6 27 = 0.222 100% = 22.2%.
8. The study design in the example was a retrospective comparative design (Gliner et al. 2009).
9. Switching the variables in the SPSS Crosstabs window would have resulted in the exact same 2 result.
10. The sample size was adequate to detect differences between the two groups because a significant difference was found p = 0.04 which is smaller than alpha = 0.05.
418
Data for Additional Computational Practice for Questions to be Graded
A retrospective comparative study examining the presence of candiduria (presence of Candida species in the urine) among 97 adults with a spinal cord injury is presented as an additional example. The differences in the use of antibiotics were investigated with the Pearson 2 test (Goetz Howard Cipher & Revankar 2010). These data are presented in Table 35-2 as a contingency table.
TABLE 35-2
CANDIDURIA AND ANTIBIOTIC USE IN ADULTS WITH SPINAL CORD INJURIES

419
EXERCISE 35Questions to Be Graded
Name: _______________________________________________________ Class: _____________________
Date: ___________________________________________________________________________________
Follow your instructor’s directions to submit your answers to the following questions for grading. Your instructor may ask you to write your answers below and submit them as a hard copy for grading. Alternatively your instructor may ask you to use the space below for notes and submit your answers online at http://evolve.elsevier.com/Grove/statistics/ under Questions to Be Graded.
1. Do the example data in Table 35-2 meet the assumptions for the Pearson 2 test? Provide a rationale for your answer.
2. Compute the 2 test. What is the 2 value?
3. Is the 2 significant at = 0.05? Specify how you arrived at your answer.
4. If using SPSS what is the exact likelihood of obtaining the 2 value at least as extreme as or as close to the one that was actually observed assuming that the null hypothesis is true?
420
5. Using the numbers in the contingency table calculate the percentage of antibiotic users who tested positive for candiduria.
6. Using the numbers in the contingency table calculate the percentage of non-antibiotic users who tested positive for candiduria.
7. Using the numbers in the contingency table calculate the percentage of veterans with candiduria who had a history of antibiotic use.
8. Using the numbers in the contingency table calculate the percentage of veterans with candiduria who had no history of antibiotic use.
9. Write your interpretation of the results as you would in an APA-formatted journal.
10. Was the sample size adequate to detect differences between the two groups in this example? Provide a rationale for your answer.
(Grove 409-420)
Grove Susan K. Daisha Cipher. Statistics for Nursing Research: A Workbook for Evidence-Based Practice 2nd Edition. Saunders 022016. VitalBook file.
The citation provided is a guideline. Please check each citation for accuracy before use.
Each exercise has 10 questions at the end which says Questions to be graded.I need those questions to be answer.

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