Question

in a large high school, 18% of sophomores and 67% of seniors have part - time jobs. suppose random samples of 32 sophomores and 31 seniors from this high school are asked if they have a part - time job. let \\( \hat { p } _ { under } \\) and \\( \hat { p } _ { upper } \\) be the sample proportions of sophomores and seniors, respectively, who have part - time jobs.
which of the following is the correct shape and justification of the sampling distribution of \\( \hat { p } _ { under } - \hat { p } _ { upper } \\) ?
bimodal because one population proportion is centered at 0.18, while the other is centered at 0.67
approximately normal because the expected numbers of successes and failures for each sample are all at least 10
not approximately normal because the expected numbers of successes and failures for each sample are all at least 10
not approximately normal because the expected numbers of successes and failures for the sophomores group are not both at least 10

Explanation:

Step1: Check for sophomores

For sophomores, \(n_1 = 32\), \(p_1=0.18\).
Expected number of successes \(n_1p_1=32\times0.18 = 5.76\)
Expected number of failures \(n_1(1 - p_1)=32\times(1 - 0.18)=32\times0.82 = 26.24\)

Step2: Check for seniors

For seniors, \(n_2 = 31\), \(p_2 = 0.67\)
Expected number of successes \(n_2p_2=31\times0.67=20.77\)
Expected number of failures \(n_2(1 - p_2)=31\times(1 - 0.67)=31\times0.33 = 10.23\)

Step3: Determine normality condition

The normality condition for the sampling distribution of \(\hat{p}_1-\hat{p}_2\) is that \(n_1p_1\geq10\), \(n_1(1 - p_1)\geq10\), \(n_2p_2\geq10\) and \(n_2(1 - p_2)\geq10\). Since \(n_1p_1 = 5.76<10\) (for sophomores), the sampling distribution of \(\hat{p}_{under}-\hat{p}_{upper}\) is not approximately normal.

Answer:

not approximately Normal because the expected numbers of successes and failures for the sophomores group are not both at least 10