Question

2. For a sample of n = 20 women aged 18 to 29, responses to the question “How tall would you like to be?" are recorded along with actual heights. In the sample, the mean desired height is 66.7 inches, the mean actual height is 64.9 inches, and the sample mean difference (desired - actual) is 1.8 inches. The sample standard deviation of the differences is 2.1 inches. Researchers hypothesize that, on average, women desire to be taller than they actually are.

1. State the null and alternative hypotheses.
   1. H0:
   2. Ha:
2. What assumptions do we need in order for the conditions of the hypothesis test procedure to be met since the sample size is 20?
3. Assuming necessary conditions are met, carry out the remaining steps of the hypothesis test (test statistic, p-value, decision, and conclusion in context).

Answer 1

x̅ = 66.7, s = 2.1, n = 20

a) Null and Alternative hypothesis:

Ho : µ = 64.9

H1 : µ > 64.9

b) since the sample size is 20, we need to assume that distribution is normal.

c) Test statistic:

t = (x̅- µ)/(s/√n) = (66.7 - 64.9)/(2.1/√20) = 3.8333

df = n-1 = 19

Critical value :

Right tailed critical value, t-crit = ABS(T.INV(0.05, 19)) = 1.729

Reject Ho if t > 1.729

p-value :

Right tailed p-value = T.DIST.RT(3.8333, 19) = 0.0006

Decision:

p-value < α, Reject the null hypothesis

There is enough evidence to conclude that women desire to be taller than they actually are at 0.05 significance level.