Question

2. Ron Littlefield is considering a campaign for mayor of Chattanooga. He will enter the race if there is evidence to suggest that fewer than p < 0.40 of all residents are satisfied with the local government. A random sample of n = 375 residents out of an estimated population of N = 2,560,000 was obtained, and x = 127 indicated that they were satisfied with the local government, or ?̂ = 0.339. Conduct the appropriate hypothesis test to determine if this politician will enter the race for mayor. Use ?=0.01.
a. Step 1: Verify the assumptions for the Distribution of the Sample Proportion ?̂ (3 pts)
• Sample is random
• Distribution is normally distributed, if n?0(1- ?0) ≥ 10
• n ≤ 0.05 of N
b. Step 2: State the null and alternative hypotheses (1 pt.):
c. Step 3: Determine the level of significance, α (1 pt.):
d. Step 4a: Calculate the test statistic (2 pts):
?0= ? ̂ − ?0√?0(1−?0)?
e. Step 4b: Determine the p-value of the test statistic (1 pt.):
f. Step 5: Compare the p-value of the test statistic to the alpha level, and decide whether to reject or retain Ho (1 pt.):
g. Step 6: State the conclusion of the hypothesis test in a full sentence (1 pt.):

Answer 1

Given that, N = 2560000

n = 375 and x = 127

a) sample is random.

np_{​​​​​0}​​​​(1-p₀) = 375 * (0.40) * (1 - 0.40) = 90  ≥ 10

and 0.05N = 0.05 * 2560000 = 128000

=>  n ≤ 0.05 of N

All assumptions are satisfied.

b) The null and alternative hypotheses are,

c) level of significance = 0.01

d) Test statistic is,

=> Z_{​​​​​​0}​​​ = -2.41

e) p-value = P(Z < -2.41) = 0.0080

=> P-value = 0.0080

​​​​​​f) Since, p-value = 0.0080 < 0.01

we reject the null hypothesis (H_{​​​​​​0}).

g) There is sufficient evidence to to suggest that fewer than p < 0.40 of all residents are satisfied with the local government.