Question

do pregnant women give birth the week of their due date? you are a researcher who wants to estimate the population proportion of all pregnant women who gave birth the week of their due date, so you will select a random sample of 40 women who have recently given birth. follow the steps below to construct a 90% confidence interval for the population proportion of all pregnant women who gave birth the week of their due date. (if necessary, consult a list of formulas.) (a) click on \take sample\ to see the results from the random sample. table with rows: \gave birth the week of due date\ (number: 18, proportion: 0.45), \did not give birth the week of due date\ (number: 22, proportion: 0.55) enter the values of the sample size, the point estimate of the population proportion, and the critical value you need for your 90% confidence interval. (choose the correct critical value from the table of critical values provided.) when you are done, select \compute\. input fields: sample size, point estimate, critical value; output fields: standard error, margin of error, 90% confidence interval table: confidence level (99%, 95%, 90%) and critical value ($z_{0.005}=2.576$, $z_{0.025}=1.960$, $z_{0.050}=1.645$) (b) based on your sample, enter the lower and upper limits to graph the 90% confidence interval for the population proportion of all pregnant women who gave birth the week of their due date. graph with 90% confidence interval axis from 0.000 to 1.000

Explanation:

Part (a)

Step 1: Determine the Sample Size

The sample consists of 40 women, so the sample size \( n = 40 \).

Step 2: Find the Point Estimate

The point estimate of the population proportion \( \hat{p} \) is the proportion of women who gave birth the week of their due date, which is given as 0.45.

Step 3: Identify the Critical Value

For a 90% confidence interval, the critical value \( z_{\alpha/2} \) is \( z_{0.050} = 1.645 \) (from the table provided).

Part (b)

Step 1: Calculate the Standard Error

The formula for the standard error (SE) of a proportion is:
\[
SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
\]
Substituting \( \hat{p} = 0.45 \), \( 1 - \hat{p} = 0.55 \), and \( n = 40 \):
\[
SE = \sqrt{\frac{0.45 \times 0.55}{40}} = \sqrt{\frac{0.2475}{40}} = \sqrt{0.0061875} \approx 0.07866
\]

Step 2: Calculate the Margin of Error (ME)

The margin of error is given by:
\[
ME = z_{\alpha/2} \times SE
\]
Substituting \( z_{\alpha/2} = 1.645 \) and \( SE \approx 0.07866 \):
\[
ME = 1.645 \times 0.07866 \approx 0.1294
\]

Step 3: Calculate the Confidence Interval

The 90% confidence interval for the population proportion \( p \) is:
\[
\hat{p} - ME < p < \hat{p} + ME
\]
Substituting \( \hat{p} = 0.45 \) and \( ME \approx 0.1294 \):
\[
0.45 - 0.1294 < p < 0.45 + 0.1294
\]
\[
0.3206 < p < 0.5794
\]

Final Answers

Part (a)

• Sample size: \( \boldsymbol{40} \)
• Point estimate: \( \boldsymbol{0.45} \)
• Critical value: \( \boldsymbol{1.645} \)

Part (b)

• Lower limit: \( \boldsymbol{0.3206} \) (or approximately 0.321)
• Upper limit: \( \boldsymbol{0.5794} \) (or approximately 0.579)

Answer:

Part (a)

Step 1: Determine the Sample Size

The sample consists of 40 women, so the sample size \( n = 40 \).

Step 2: Find the Point Estimate

The point estimate of the population proportion \( \hat{p} \) is the proportion of women who gave birth the week of their due date, which is given as 0.45.

Step 3: Identify the Critical Value

For a 90% confidence interval, the critical value \( z_{\alpha/2} \) is \( z_{0.050} = 1.645 \) (from the table provided).

Part (b)

Step 1: Calculate the Standard Error

The formula for the standard error (SE) of a proportion is:
\[
SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
\]
Substituting \( \hat{p} = 0.45 \), \( 1 - \hat{p} = 0.55 \), and \( n = 40 \):
\[
SE = \sqrt{\frac{0.45 \times 0.55}{40}} = \sqrt{\frac{0.2475}{40}} = \sqrt{0.0061875} \approx 0.07866
\]

Step 2: Calculate the Margin of Error (ME)

The margin of error is given by:
\[
ME = z_{\alpha/2} \times SE
\]
Substituting \( z_{\alpha/2} = 1.645 \) and \( SE \approx 0.07866 \):
\[
ME = 1.645 \times 0.07866 \approx 0.1294
\]

Step 3: Calculate the Confidence Interval

The 90% confidence interval for the population proportion \( p \) is:
\[
\hat{p} - ME < p < \hat{p} + ME
\]
Substituting \( \hat{p} = 0.45 \) and \( ME \approx 0.1294 \):
\[
0.45 - 0.1294 < p < 0.45 + 0.1294
\]
\[
0.3206 < p < 0.5794
\]

Final Answers

Part (a)

• Sample size: \( \boldsymbol{40} \)
• Point estimate: \( \boldsymbol{0.45} \)
• Critical value: \( \boldsymbol{1.645} \)

Part (b)

• Lower limit: \( \boldsymbol{0.3206} \) (or approximately 0.321)
• Upper limit: \( \boldsymbol{0.5794} \) (or approximately 0.579)