Question

A large study involving over 20,000 individuals shows that the mean percentage intake of kilocalories from fat was 39% with a range from 6% to 72%. A small sample study was conducted at a university hospital to determine if the mean intake of patients at that hospital was different from 39%. A sample of 15 patients had a mean intake of 40.8% with a standard deviation equal to 6.5%. Assume that the sample is from a normally distributed population. Use a 5% level of significance.

a). State your hypotheses for the test:

b). Determine the critical value or the p-value.

c). From the question determine the important data with its corresponding notations.

d). Calculate the test statistic for this scenario.

e). Make the necessary decisions and state your conclusions.

Answer 1

Below are the null and alternative Hypothesis,
Null Hypothesis, H0: μ = 39
Alternative Hypothesis, Ha: μ ≠ 39

Rejection Region
This is two tailed test, for α = 0.05 and df = 14
Critical value of t are -2.145 and 2.145.
Hence reject H0 if t < -2.145 or t > 2.145

xbar = 40.8
mu = 39, n = 15 and s = 6.5

Test statistic,
t = (xbar - mu)/(s/sqrt(n))
t = (40.8 - 39)/(6.5/sqrt(15))
t = 1.073

P-value Approach
P-value = 0.3014

As P-value >= 0.05, fail to reject null hypothesis.

There is not sufficient evidence to conclude that the percentage is different than 39