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Math  /  Data & Statistics

Questionwww-awualeks.com Content Aleks Blackboard Content Session Clo... Content Jonathan V-. \mathrm{V}_{\text {-. }} Timed Out knicks - Go... NFL Footba... Homework * 5: 9(1,3,4,5) 14(1,2) Question 10 of 30 (1 point) I Question Attempt: 1 of 3
Spam: A researcher reported that 71.8%71.8 \% of all email sent in a recent month was spam. A system manager at a large corporation believes that the percentage at his company may be 77%77 \%. He examines a random sample of 500 emails received at an email server, and finds that 364 of the messages are spam. Can you conclude that the percentage of emails that are spam differs from 77%77 \% ? Use both α=0.01\alpha=0.01 and α=0.05\alpha=0.05 levels of significance and the PP-value method with the TI-84 Plus calculator.
Part 1 of 5 (a) State the appropriate null and alternate hypotheses. H0:□H1:□\begin{array}{l} H_{0}: \square \\ H_{1}: \square \end{array}
This hypotheses test is a left-tailed ∇\quad \nabla test.
Correct Answer: H0:p=0.77H1:p≠0.77\begin{array}{l} H_{0}: p=0.77 \\ H_{1}: p \neq 0.77 \end{array}
This hypotheses test is a two-tailed test.
Part: 1/51 / 5
Part 2 of 5 (b) Compute the value of the test statistic. Round the answer to at least two decimal places. z=□z=\square

Studdy Solution

STEP 1

1. We are conducting a hypothesis test for a proportion.
2. The null hypothesis is H0:p=0.77 H_0: p = 0.77 .
3. The alternative hypothesis is H1:p≠0.77 H_1: p \neq 0.77 .
4. The sample size is n=500 n = 500 .
5. The number of spam emails in the sample is x=364 x = 364 .
6. The sample proportion is calculated as p^=xn \hat{p} = \frac{x}{n} .

STEP 2

1. Calculate the sample proportion.
2. Compute the standard error of the sample proportion.
3. Calculate the test statistic using the standard normal distribution.

STEP 3

Calculate the sample proportion:
p^=364500=0.728 \hat{p} = \frac{364}{500} = 0.728

STEP 4

Compute the standard error of the sample proportion using the formula:
SE=p0(1−p0)n \text{SE} = \sqrt{\frac{p_0(1-p_0)}{n}}
where p0=0.77 p_0 = 0.77 .
SE=0.77×(1−0.77)500=0.77×0.23500 \text{SE} = \sqrt{\frac{0.77 \times (1 - 0.77)}{500}} = \sqrt{\frac{0.77 \times 0.23}{500}}
SE=0.1771500=0.0003542≈0.0188 \text{SE} = \sqrt{\frac{0.1771}{500}} = \sqrt{0.0003542} \approx 0.0188

STEP 5

Calculate the test statistic z z using the formula:
z=p^−p0SE z = \frac{\hat{p} - p_0}{\text{SE}}
z=0.728−0.770.0188 z = \frac{0.728 - 0.77}{0.0188}
z=−0.0420.0188≈−2.23 z = \frac{-0.042}{0.0188} \approx -2.23
The value of the test statistic is:
z≈−2.23 z \approx -2.23

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