Question

PU Math 80 - Statistics Central Limit Theorem for Sample Proportions Handout 5.2 2 ES school 0.5. schools face an ongoing challenge: as students get closer to high school graduation, their enthusiasm for school falls. A Gallup poll in June 2017 has discovered that fifth graders are most engaged with school, while 11th graders are least. The latest Gallup poll showed that only 32% of 11th graders are engaged in school. Assume this value represents the population proportion. Suppose we randomly sample 150 11" graders in the U.S., and ask them about their engagement in school. 0.32 A Samples of size 150 are taken, find the mean, we and standard error, da of the resulting sampling distribution of sample, proportions, p. P-307 -0.37 M 150*0.37 B What does this sampling distribution represent? C Based on the mean and standard error, label the values of a through g on the normal distribution below. The values are one standard deviation apart a b c d e f g D Use the empirical rule to find the interval centered at p that contains approximately 99.7% of all sample proportions. E What sample proportions would you consider unusual? Give two examples, one on each side of the mean.

Answer 1

2. A) The mean of sampling distribution = Mean of the population from which the samples are collected

p = 0.32
Standard error = √(pq/n) = √(0.32*0.68/150) = 0.038

B) This sampling distribution represent the probability distribution of the mean values for all the possible samples of size 150

C) Point a: p - 3*SE = 0.32 - 3*0.038 = 0.206
b: p - 2*SE = 0.32 - 2*0.038 = 0.244
c: p - 3*SE = 0.32 - 1*0.038 = 0.282
d: p = Mean = 0.320
e: p + 1*SE = 0.32 + 1*0.038 = 0.358
f: p + 2*SE = 0.32 + 2*0.038 = 0.396
g: p + 3*SE = 0.32 + 3*0.038 = 0.434

D) The empirical rule states that the 99.7% of all sample proportions will fall around three standard deviations around the mean.

Interval: (a, g) = (0.206, 0.434)

E) The sample proportions we will consider unusual are which fall more than three standard deviations from the mean.

Unusual: p < 0.206 OR p>0.434

Examples : 0.2 and 0.45