Question

An investigator conducts a study to investigate whether there is a difference in mean PEF in children with chronic bronchitis as compared to those without. Data on PEF are collected and summarized below. Based on the data, is there statistical evidence of a lower mean PEF in children with chronic bronchitis as compared to those without? Run the appropriate test at α=0.05. Make sure to 1. give the null and alternative hypotheses, 2. calculate the test statistic, and 3. give the conclusion (including a comparison to alpha or the critical value).

Group                               Number of children           Mean PEF        Std Dev PEF

Chronic Bronchitis                       42 265 65

No Chronic Bronchitis                 40 319 70

a) Hypotheses

                 HO: vs. HA  

b) Calculate S²p=

c) Compute Test statistic  T=

d) P-value

e) Conclusion

             Accept H0                                               Reject H0

f) Calculate a 95% confidence interval for mean difference  

Answer 1

Group	Number of children	Mean PEF	Std Dev PEF
Chronic Bronchitis	42	265	65
No Chronic Bronchitis	40	319	70

The provided sample means are shown below:

Also, the provided sample standard deviations are:

and the sample sizes are

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

​

​

This corresponds to a left-tailed test, for which a t-test for two population means, with two independent samples, with unknown population standard deviations will be used.

(2)

Since the population variances are assumed to be equal, we need to compute the pooled standard deviation, as follows:

c)

Test Statistics

Since it is assumed that the population variances are equal, the t-statistic is computed as follows:

d)

The p-value is p = 0.0003

e)

Reject H0

Since it is observed that t=−3.622<tc​=−1.664, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p=0.0003, and since p=0.0003<0.05, it is concluded that the null hypothesis is rejected.

f)

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