normal condition for sampling proportions - According to the admissions director of a certain college, approximately 9 % of the 150 freshmen admitted had applied using the early decision option. Suppose that we took random samples of 15 freshmen from this population and computed the proportion of freshmen in each sample who had applied using the early decision option. We can assume the admissions director's claim is true. What will be the shape of the sampling distribution of p?

According to the admissions director of a certain college, approximately 9 % of the 150 freshmen admitted had applied using the early decision option. Suppose that we took random samples of 15 freshmen from this population and computed the proportion of freshmen in each sample who had applied using the early decision option. We can assume the admissions director's claim is true. What will be the shape of the sampling distribution of p?

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normal condition for sampling proportions ap statistics approximately 11% of the 250 freshmen admitted had applied using the early decision option

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Sangeetha Pulapaka (last edited 1 year ago) (Expert, Educator @Qalaxia)

We are looking at which conditions does the sampling distributions of the sample proportions, looks, skewed to the left, skewed to the right, approximately normal, or uniform.

Our sample size n = 15, and p = 0.09

Expected successes: np = 15(0.09) = 1.35

Expected failures ; n(1-p) = 15 ( 1 - 0.09) = 15(0.91) = 13.65

Since np\lt 10, and n(1-p) \geq 10, the shape of the sampling proportion is not normal.

Since we only expect about 1 freshmen who applied using the early decision option in the sample, which is less than 10, the sample proportions will not be normally distributed.

Now to find in which direction the shape of the sampling distribution is skewed.

We would expect only a right-skewed sampling distribution when the population proportion is so extreme relative to the sample size that there are fewer than 10 expected successes per sample. So, since our expected successes is less than 10, we have a right-skewed sampling distribution.

The sampling distribution of \widehat{p} will be skewed to the right.

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