Chapter 15, Problem 1E

Website An investment company is planning to upgrade the mobile access to their website, but they’d like to know the proportion of their customers who access it from their smartphones. They draw a random sample of 200 from customers who recently logged in and check their IP address. Suppose that the true proportion of smartphone users is 36%.

1. a) What would you expect the shape of the sampling distribution for the sample proportion to be?
2. b) What would be the mean of this sampling distribution?
3. c) What would be the standard deviation of the sampling distribution?

To determine

Explain the shape of the sampling distribution for the sample proportion.

Answer to Problem 1E

The shape of the sampling distribution for the sample proportion is unimodal and symmetric.

Explanation of Solution

Given info:

A random sample of 200 is drawn from customers who recently logged in and checked their IP address and the smartphone users true proportion is 36%.

Justification:

Sampling distribution for a proportion:

For all possible samples the distribution of proportion is called the sampling distribution of proportion

• Samples should be independent.
• Sample size should be large.
• The sampling distribution of $\widehat{p}$ is formed by normal model with mean p and standard deviation of $\widehat{p}$ is, $SD\left(\widehat{p}\right)=\sqrt{\frac{pq}{n}}$ where $q=1-p$.

Here random sample is drawn by considering 200 investors as a small portion of their customer. Therefore, the sampling distribution for the proportion of 200 investors that use smartphones would be unimodal and symmetrical.

To determine

Find the mean of this sampling distribution.

Answer to Problem 1E

The mean of this sampling distribution is 0.36.

Explanation of Solution

Justification:

From part (a), the mean for the sampling distribution of proportion is p.

Here, the true proportion smartphone users, is 36% that is, 0.36.

Thus, the mean of this sampling distribution is 0.36.

To determine

Find the standard deviation of the sampling distribution.

Answer to Problem 1E

The standard deviation of the sampling distribution is 0.034.

Explanation of Solution

Calculation:

From part (a), the standard deviation of $\widehat{p}$ is, $SD\left(\widehat{p}\right)=\sqrt{\frac{pq}{n}}$.

Here proportion p is 0.36 and q is $0.64=\left(1-0.36\right)$ and total sample is 200.

Substitute p as 0.36, q as 0.64 and n as 200 in the formula,

$\begin{array}{c}SD\left(\widehat{p}\right)=\sqrt{\frac{0.36\left(0.64\right)}{200}}\\ =\sqrt{\frac{0.2304}{200}}\\ =\sqrt{0.001152}\\ \approx 0.034\end{array}$

Thus, the standard deviation of the sampling distribution is 0.034.