In a sample of 49 adolescents who served as the subjects in an immunologic study, one variable of interest was the diameter of a skin test reaction to an allergen. The sample mean and standard deviation were 21 and 11 mm erythema, respectively.

a. Use a z - or t-distribution? Why?

b. One-sided or two-sided test? Why?

c. Can it be concluded from these data that the population mean is less than 24 mm erythema?

d. What is the range on the p-value?

Step-by-step explanation:

a) We would use a t distribution because the population standard deviation is unknown.

b) it is a one sided test because we are trying to determine if the population mean is less than 24 mm erythema. The lesser than means that it is a left tailed test.

c) To determine the p value, we would apply the formula,

t = (x - µ) / (s/√n)

Where

x = sample mean = 21

µ = population mean = 24

s = samples standard deviation = 11

n = number of samples = 49

t = (21 - 24) / (11/√49) = - 1.91

Since n = 49

Degrees of freedom, df = n - 1 = 49 - 1 = 48

We would determine the p value using the t test calculator. It becomes

p = 0.031

Assuming alpha = 0.05, then

Since alpha, 0.05 > than the p value, 0.031, then we would reject the null hypothesis. Therefore, At a 5% level of significance, it can be concluded from these data that the population mean is less than 24 mm erythema.

d) p value = 0.031