Math / Data & Statistics

QuestionQuestion 6, 9.1.15 HW Score: $55.56 \%, 5$ of 9 points .1
Construct a $90 \%$ confidence interval of the population proportion using the given information. $x=125, n=250$
Click here to view the table of critical values.
The lower bound is $\square$ The upper bound is $\square$ (Round to three decimal places as needed. Table of critical values \begin{tabular}{ccc} \begin{tabular}{c} Level of Confidence, \\ $(\mathbf{1}-\alpha) \cdot \mathbf{1 0 0} \%$ \end{tabular} & Area in Each Tail, $\frac{\boldsymbol{\alpha}}{\mathbf{2}}$ & Critical Value, $\boldsymbol{z}_{\frac{\boldsymbol{\alpha}}{\mathbf{2}}}$ \\ \hline $90 \%$ & 0.05 & 1.645 \\ \hline $95 \%$ & 0.025 & 1.96 \\ \hline $99 \%$ & 0.005 & 2.575 \\ \hline \end{tabular} Print Done

Studdy Solution

STEP 1

1. We are constructing a confidence interval for a population proportion.
2. The sample size $n = 250$ and the number of successes $x = 125$ .
3. The level of confidence is $90\%$ , corresponding to a critical value $z_{\frac{\alpha}{2}} = 1.645$ .

STEP 2

1. Calculate the sample proportion.
2. Determine the standard error of the sample proportion.
3. Calculate the margin of error.
4. Construct the confidence interval.

STEP 3

Calculate the sample proportion $\hat{p}$ :
$\hat{p} = \frac{x}{n} = \frac{125}{250} = 0.5$

STEP 4

Calculate the standard error (SE) of the sample proportion:
$SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} = \sqrt{\frac{0.5 \times (1 - 0.5)}{250}} = \sqrt{\frac{0.25}{250}} = \sqrt{0.001} = 0.0316$

STEP 5

Calculate the margin of error (ME):
$ME = z_{\frac{\alpha}{2}} \times SE = 1.645 \times 0.0316 = 0.052$

STEP 6

Construct the confidence interval:
Lower bound:
$\hat{p} - ME = 0.5 - 0.052 = 0.448$
Upper bound:
$\hat{p} + ME = 0.5 + 0.052 = 0.552$
The lower bound is $\boxed{0.448}$ and the upper bound is $\boxed{0.552}$ .

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Studdy Solution

STEP 1

1. We are constructing a confidence interval for a population proportion.
2. The sample size $n = 250$ and the number of successes $x = 125$ .
3. The level of confidence is $90\%$ , corresponding to a critical value $z_{\frac{\alpha}{2}} = 1.645$ .

STEP 2

1. Calculate the sample proportion.
2. Determine the standard error of the sample proportion.
3. Calculate the margin of error.
4. Construct the confidence interval.

STEP 3

Calculate the sample proportion $\hat{p}$ :
$\hat{p} = \frac{x}{n} = \frac{125}{250} = 0.5$

STEP 4

Calculate the standard error (SE) of the sample proportion:
$SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} = \sqrt{\frac{0.5 \times (1 - 0.5)}{250}} = \sqrt{\frac{0.25}{250}} = \sqrt{0.001} = 0.0316$

STEP 5

Calculate the margin of error (ME):
$ME = z_{\frac{\alpha}{2}} \times SE = 1.645 \times 0.0316 = 0.052$

STEP 6

Construct the confidence interval:
Lower bound:
$\hat{p} - ME = 0.5 - 0.052 = 0.448$
Upper bound:
$\hat{p} + ME = 0.5 + 0.052 = 0.552$
The lower bound is $\boxed{0.448}$ and the upper bound is $\boxed{0.552}$ .

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Studdy Solution

STEP 1

1. We are constructing a confidence interval for a population proportion.2. The sample size n=250 n = 250 n=250 and the number of successes x=125 x = 125 x=125.3. The level of confidence is 90% 90\% 90%, corresponding to a critical value zα2=1.645 z_{\frac{\alpha}{2}} = 1.645 z2α​​=1.645.

STEP 2

1. Calculate the sample proportion.2. Determine the standard error of the sample proportion.3. Calculate the margin of error.4. Construct the confidence interval.

STEP 3

Calculate the sample proportion p^ \hat{p} p^​:p^=xn=125250=0.5\hat{p} = \frac{x}{n} = \frac{125}{250} = 0.5p^​=nx​=250125​=0.5

STEP 4

STEP 5

Calculate the margin of error (ME):ME=zα2×SE=1.645×0.0316=0.052ME = z_{\frac{\alpha}{2}} \times SE = 1.645 \times 0.0316 = 0.052ME=z2α​​×SE=1.645×0.0316=0.052

STEP 6

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1. We are constructing a confidence interval for a population proportion.
2. The sample size $n = 250$ and the number of successes $x = 125$ .
3. The level of confidence is $90\%$ , corresponding to a critical value $z_{\frac{\alpha}{2}} = 1.645$ .

1. Calculate the sample proportion.
2. Determine the standard error of the sample proportion.
3. Calculate the margin of error.
4. Construct the confidence interval.

Calculate the sample proportion $\hat{p}$ :
$\hat{p} = \frac{x}{n} = \frac{125}{250} = 0.5$

Calculate the margin of error (ME):
$ME = z_{\frac{\alpha}{2}} \times SE = 1.645 \times 0.0316 = 0.052$