Math / Data & Statistics

Questionothesis Tests for a Question 3, 10.2.17-T HW Score: $35.88 \%, 2.51$ of 7 points Part 1 of 5 Points: 0 of 1 Save
Researchers selected 825 patients at random among those who take a certain widely-used prescription drug daily. In a clinical trial, 23 out of the 825 patients complained of flulike symptoms. Suppose that it is known that $2.5 \%$ of patients taking competing drugs complain of flulike symptoms. Is there sufficient evidence to conclude that more than $2.5 \%$ of this drug's users expenience flulike symptoms as a side effect at the $\alpha=0.01$ level of significance?
Because $\mathrm{np}_{0}\left(1-p_{0}\right)=$ $\square$ $\square$ 10 , the sample size is $\square$ $5 \%$ of the population size, and the patients in the sample $\square$ selected at random, all of the requirements for testing the hypothesis $\square$ satisfied. (Round to one decimal place as needed.)

Studdy Solution

STEP 1

1. The null hypothesis ( $H_0$ ) is that the proportion of patients experiencing flulike symptoms is $p_0 = 0.025$ .
2. The alternative hypothesis ( $H_a$ ) is that the proportion is greater than $0.025$ .
3. The sample size is $n = 825$ .
4. The number of patients with flulike symptoms is $x = 23$ .
5. The significance level is $\alpha = 0.01$ .

STEP 2

1. Check the conditions for hypothesis testing.
2. Calculate the test statistic.
3. Determine the critical value and compare it with the test statistic.
4. Make a decision based on the comparison.

STEP 3

Check the condition $\mathrm{np}_{0}(1-p_{0}) \geq 10$ .
Calculate $\mathrm{np}_{0}(1-p_{0})$ :
$\mathrm{np}_{0}(1-p_{0}) = 825 \times 0.025 \times (1 - 0.025) = 825 \times 0.025 \times 0.975$
$= 20.0625$
Since $20.0625 \geq 10$ , this condition is satisfied.

STEP 4

Check if the sample size is less than $5\%$ of the population size. Assume the population size is large enough that this condition is satisfied.

STEP 5

Verify that the sample is selected at random. This is given in the problem statement.

STEP 6

Calculate the sample proportion $\hat{p}$ :
$\hat{p} = \frac{x}{n} = \frac{23}{825} \approx 0.0279$
Calculate the standard error $SE$ :
$SE = \sqrt{\frac{p_0(1-p_0)}{n}} = \sqrt{\frac{0.025 \times 0.975}{825}}$
$\approx 0.0054$
Calculate the test statistic $z$ :
$z = \frac{\hat{p} - p_0}{SE} = \frac{0.0279 - 0.025}{0.0054}$
$\approx 0.537$

STEP 7

Determine the critical value for a one-tailed test at $\alpha = 0.01$ . The critical value $z_{\alpha}$ is approximately $2.33$ .

STEP 8

Compare the test statistic to the critical value:
Since $0.537 < 2.33$ , we do not reject the null hypothesis.

STEP 9

Make a decision:
There is not sufficient evidence to conclude that more than $2.5\%$ of this drug's users experience flulike symptoms as a side effect at the $\alpha = 0.01$ level of significance.

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Studdy Solution

STEP 1

STEP 2

1. Check the conditions for hypothesis testing.
2. Calculate the test statistic.
3. Determine the critical value and compare it with the test statistic.
4. Make a decision based on the comparison.

STEP 3

Check the condition $\mathrm{np}_{0}(1-p_{0}) \geq 10$ .
Calculate $\mathrm{np}_{0}(1-p_{0})$ :
$\mathrm{np}_{0}(1-p_{0}) = 825 \times 0.025 \times (1 - 0.025) = 825 \times 0.025 \times 0.975$
$= 20.0625$
Since $20.0625 \geq 10$ , this condition is satisfied.

STEP 4

Check if the sample size is less than $5\%$ of the population size. Assume the population size is large enough that this condition is satisfied.

STEP 5

Verify that the sample is selected at random. This is given in the problem statement.

STEP 6

STEP 7

Determine the critical value for a one-tailed test at $\alpha = 0.01$ . The critical value $z_{\alpha}$ is approximately $2.33$ .

STEP 8

Compare the test statistic to the critical value:
Since $0.537 < 2.33$ , we do not reject the null hypothesis.

STEP 9

Make a decision:
There is not sufficient evidence to conclude that more than $2.5\%$ of this drug's users experience flulike symptoms as a side effect at the $\alpha = 0.01$ level of significance.

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Studdy Solution

STEP 1

STEP 2

1. Check the conditions for hypothesis testing.2. Calculate the test statistic.3. Determine the critical value and compare it with the test statistic.4. Make a decision based on the comparison.

STEP 3

STEP 4

Check if the sample size is less than 5%5\%5% of the population size. Assume the population size is large enough that this condition is satisfied.

STEP 5

Verify that the sample is selected at random. This is given in the problem statement.

STEP 6

STEP 7

Determine the critical value for a one-tailed test at α=0.01\alpha = 0.01α=0.01. The critical value zαz_{\alpha}zα​ is approximately 2.332.332.33.

STEP 8

Compare the test statistic to the critical value:Since 0.537<2.330.537 < 2.330.537<2.33, we do not reject the null hypothesis.

STEP 9

Make a decision:There is not sufficient evidence to conclude that more than 2.5%2.5\%2.5% of this drug's users experience flulike symptoms as a side effect at the α=0.01\alpha = 0.01α=0.01 level of significance.

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Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. Check the conditions for hypothesis testing.
2. Calculate the test statistic.
3. Determine the critical value and compare it with the test statistic.
4. Make a decision based on the comparison.

Check if the sample size is less than $5\%$ of the population size. Assume the population size is large enough that this condition is satisfied.

Determine the critical value for a one-tailed test at $\alpha = 0.01$ . The critical value $z_{\alpha}$ is approximately $2.33$ .

Compare the test statistic to the critical value:
Since $0.537 < 2.33$ , we do not reject the null hypothesis.

Make a decision:
There is not sufficient evidence to conclude that more than $2.5\%$ of this drug's users experience flulike symptoms as a side effect at the $\alpha = 0.01$ level of significance.