Thirty-three percent of all Americans drink bottled water more than once a week (Natural Resources Defense Council, December 4, 2015). Suppose you have been hired by the Natural Resources Defense Council to investigate bottled water consumption in St. Paul. You plan to select a sample of St. Paulites to estimate the proportion who drink bottled water more than once a week. Assume the population proportion of St. Paulites who drink bottled water more than once a week is 0.33, the same as the overall proportion of Americans who drink bottled water more than once a week. Use the [tex]$z$[/tex]-table.

a. Suppose you select a sample of 540 St. Paulites. Show the sampling distribution of [tex]$\bar{p}$[/tex] (to 4 decimals).

[tex]\[
\begin{array}{l}
E(\bar{p})=0.33 \\
\sigma_{\bar{p}}=0.0202
\end{array}
\][/tex]

b. Based upon a sample of 540 St. Paulites, what is the probability that the sample proportion will be within 0.02 of the population proportion (to 4 decimals).

[tex]\[
\text{probability} = 0.9998
\][/tex]

c. Suppose you select a sample of 240 St. Paulites. Show the sampling distribution of [tex]$\bar{p}$[/tex] (to 4 decimals).

[tex]\[
\begin{array}{l}
E(\bar{p})=0.33 \\
\sigma_{\bar{p}}=0.0296
\end{array}
\][/tex]

d. Based upon a smaller sample of only 240 St. Paulites, what is the probability that the sample proportion will be within 0.02 of the population proportion (to 4 decimals).

[tex]\[
\text{probability} = 0.8340
\][/tex]



Answer :

Let's work through each part of the question step by step. The problem involves understanding the statistical concepts of sampling distribution, calculating the standard error, and finding probabilities using the normal distribution.

### Part (a)
Sampling distribution for a sample of 540 St. Paulites:

1. Expected value of [tex]\(\bar{p}\)[/tex]: The expected value [tex]\(E(\bar{p})\)[/tex] is the same as the population proportion [tex]\(p\)[/tex]. Given that [tex]\(p = 0.33\)[/tex]:
[tex]\[ E(\bar{p}) = 0.33 \][/tex]

2. Standard error of [tex]\(\bar{p}\)[/tex]: The standard error [tex]\(\sigma_{\bar{p}}\)[/tex] for the sample proportion is calculated using the formula:
[tex]\[ \sigma_{\bar{p}} = \sqrt{\frac{p(1 - p)}{n}} \][/tex]
For a sample size [tex]\(n = 540\)[/tex]:
[tex]\[ \sigma_{\bar{p}} = \sqrt{\frac{0.33 \times (1 - 0.33)}{540}} = \sqrt{\frac{0.33 \times 0.67}{540}} = 0.0202 \][/tex]

So, the sampling distribution of [tex]\(\bar{p}\)[/tex] for a sample size of 540 is:
[tex]\[ E(\bar{p}) = 0.33, \quad \sigma_{\bar{p}} = 0.0202 \][/tex]

### Part (b)
Probability that the sample proportion will be within 0.02 of the population proportion for a sample of 540:

1. Margin of error: We are asked to find the probability that the sample proportion is within 0.02 of the population proportion [tex]\(0.33\)[/tex].

2. Z-score calculation: The Z-score for the margin can be calculated as follows:
[tex]\[ Z = \frac{margin}{\sigma_{\bar{p}}} = \frac{0.02}{0.0202} = 0.990 \][/tex]

3. Using the standard normal distribution table: We need to find the probability that Z is between [tex]\(-0.990\)[/tex] and [tex]\(0.990\)[/tex]:
[tex]\[ P(-0.990 \leq Z \leq 0.990) \][/tex]

From the standard normal distribution table, we find that the cumulative probability for Z = 0.990 is approximately 0.677:
[tex]\[ P(-0.990 \leq Z \leq 0.990) = 0.677 \][/tex]

### Part (c)
Sampling distribution for a sample of 240 St. Paulites:

1. Expected value of [tex]\(\bar{p}\)[/tex]: The expected value [tex]\(E(\bar{p})\)[/tex] is still the population proportion [tex]\(p\)[/tex]. Given that [tex]\(p = 0.33\)[/tex]:
[tex]\[ E(\bar{p}) = 0.33 \][/tex]

2. Standard error of [tex]\(\bar{p}\)[/tex]: The standard error [tex]\(\sigma_{\bar{p}}\)[/tex] for the sample proportion is calculated using the same formula, but with [tex]\(n = 240\)[/tex]:
[tex]\[ \sigma_{\bar{p}} = \sqrt{\frac{0.33 \times (1 - 0.33)}{240}} = \sqrt{\frac{0.33 \times 0.67}{240}} = 0.0304 \][/tex]

So, the sampling distribution of [tex]\(\bar{p}\)[/tex] for a sample size of 240 is:
[tex]\[ E(\bar{p}) = 0.33, \quad \sigma_{\bar{p}} = 0.0304 \][/tex]

### Part (d)
Probability that the sample proportion will be within 0.02 of the population proportion for a sample of 240:

1. Margin of error: We are asked to find the probability that the sample proportion is within 0.02 of the population proportion [tex]\(0.33\)[/tex].

2. Z-score calculation: The Z-score for the margin can be calculated as follows:
[tex]\[ Z = \frac{margin}{\sigma_{\bar{p}}} = \frac{0.02}{0.0304} = 0.658 \][/tex]

3. Using the standard normal distribution table: We need to find the probability that Z is between [tex]\(-0.658\)[/tex] and [tex]\(0.658\)[/tex]:
[tex]\[ P(-0.658 \leq Z \leq 0.658) \][/tex]

From the standard normal distribution table, we find that the cumulative probability for Z = 0.658 is approximately 0.4901:
[tex]\[ P(-0.658 \leq Z \leq 0.658) = 0.4901 \][/tex]

In summary, for part (d), the probability that the sample proportion will be within 0.02 of the population proportion is 0.4901.

To recap:
a. For [tex]\(n = 540\)[/tex]:
- [tex]\(E(\bar{p}) = 0.33\)[/tex]
- [tex]\(\sigma_{\bar{p}} = 0.0202\)[/tex]

b. The probability within 0.02 for [tex]\(n = 540\)[/tex] is 0.677

c. For [tex]\(n = 240\)[/tex]:
- [tex]\(E(\bar{p}) = 0.33\)[/tex]
- [tex]\(\sigma_{\bar{p}} = 0.0304\)[/tex]

d. The probability within 0.02 for [tex]\(n = 240\)[/tex] is 0.4901