SOLUTION: Peak expiratory flow (PEF) is a measure of a patient’s ability to expel air from the lungs. Patients with asthma or other respiratory conditions often have restricted PEF. The

Question 1142702: Peak expiratory flow (PEF) is a measure of a patient’s ability to expel air from the lungs. Patients with asthma or other respiratory conditions often have restricted PEF. The mean PEF for children free of asthma is 360. An investigator wants to test whether children with chronic bronchitis have restricted PEF. A sample of 48 children with chronic bronchitis are studied and their mean PEF is 297 with a standard deviation of 74. Is there statistical evidence of a lower mean PEF in children with chronic bronchitis? Run the appropriate test at alpha = 0.05.
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
since you want to test if the PEF is lower, then this would be a one tail test with alpha of .05 on the left side of the normal distribution.

the mean PEF for children free of asthma is 36.
a sample of 48 children with asthma had a mean of 297 with a sample standard deviation of 74.

since your standard deviation is from the sample, and not from the population, you would do a t-test.

at .05 alpha on the left, the critical t-score with 47 degrees of freedom would be equal to -1.68 rounded to 2 decimal places.

the t-score formula is t = (x-m)/s

t is the t-score for the sample.
x is the mean of the sample.
m is the mean of the population you are comparing against.
s is the standard error of the distribution of sample means.

s = standard deviation of the sample / square root of the sample size.
that becomes s = 74 / sqrt(48) = 10.68098 rounded to 5 decimal places.

formula becomes t = (297 - 360) / 10.68098 = -5.89335 rounded to 5 decimal places.

since the critical t-score is -1.68, then the sample t-score is way beyond this, indicating that there is an overwhelmingly high probability that the mean PEF of children with chronic asthma is less than the mean PEF of children without asthma and that the difference is not due to random variations in the mean of samples whose size is 48.

the alpha for a t-score of -5.89335 with 47 degrees of freedom is equal to .000000191735.

this is effectively equal to 0 because it's so small and it is well below the critical alpha of .05.

since your sample t-score was beyond the value of the critical t-score, then your sample alpha should have been less than the critical alpha, which it was.

this means that there is statistical evidence of a lower mean PEF in children with chronic bronchitis.

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