Math / Data & Statistics

QuestionA research study was conducted about gender differences in "sexting." The researcher belleved that the proportion of girls involved in "sexting" is less than the proportion of boys involved. The data collected in the spring of 2010 among a random sample of middle and high school students in a large school district in the southern United States is out of 2231 males, 183 particpated in sexting. Out of 2169, 156 particpated. Is the proportion of girls sending sexts less than the proportion of boys "sexting?" Test at a 1\% level of significance. a) If we use $B$ to denote the boys and $G$ to denote the girls, identify the correct alternative hypothesis. $H_{1}: p_{B} \neq p_{G}$ $H_{1}: p_{B}<p_{G}$ $H_{1}: p_{B}>p_{G}$ b) Determine the test statistic. Round to two decimals. $z=\cdot \square$ c) Find the phalue and round to 4 decimals. $p=$ $\square$ d) Make a decision. Reject the null hypothesis Fail to reject the null hypothesis e) Write the conclusion. There is sufficient evidence to support the claim that the proportion of girls who sext is less than boys. There is not sufficient evidence to support the claim that the proportion of girls who sext is less than boys.

Studdy Solution

STEP 1

1. We are conducting a hypothesis test for the difference in proportions between two independent groups (boys and girls).
2. The level of significance is $\alpha = 0.01$ .

STEP 2

1. Identify the correct alternative hypothesis.
2. Calculate the test statistic.
3. Find the p-value.
4. Make a decision based on the p-value.
5. Write the conclusion.

STEP 3

Identify the correct alternative hypothesis. The researcher believes that the proportion of girls involved in "sexting" is less than the proportion of boys. Therefore, the alternative hypothesis is:
$H_{1}: p_{B} > p_{G}$

STEP 4

Calculate the test statistic. First, find the sample proportions:
$\hat{p}_{B} = \frac{183}{2231}, \quad \hat{p}_{G} = \frac{156}{2169}$
Calculate the pooled proportion:
$\hat{p} = \frac{183 + 156}{2231 + 2169}$
Calculate the standard error:
$SE = \sqrt{\hat{p}(1 - \hat{p}) \left( \frac{1}{2231} + \frac{1}{2169} \right)}$
Calculate the test statistic:
$z = \frac{\hat{p}_{B} - \hat{p}_{G}}{SE}$

STEP 5

Find the p-value using the z-test statistic obtained in STEP_2. Since this is a one-tailed test, find the probability that $Z$ is greater than the calculated z-value.

STEP 6

Make a decision. Compare the p-value to the significance level $\alpha = 0.01$ .
- If $p \leq 0.01$ , reject the null hypothesis. - If $p > 0.01$ , fail to reject the null hypothesis.

STEP 7

Write the conclusion based on the decision made in STEP_4.
- If the null hypothesis is rejected, conclude: "There is sufficient evidence to support the claim that the proportion of girls who sext is less than boys." - If the null hypothesis is not rejected, conclude: "There is not sufficient evidence to support the claim that the proportion of girls who sext is less than boys."

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Studdy Solution

STEP 1

1. We are conducting a hypothesis test for the difference in proportions between two independent groups (boys and girls).
2. The level of significance is $\alpha = 0.01$ .

STEP 2

1. Identify the correct alternative hypothesis.
2. Calculate the test statistic.
3. Find the p-value.
4. Make a decision based on the p-value.
5. Write the conclusion.

STEP 3

Identify the correct alternative hypothesis. The researcher believes that the proportion of girls involved in "sexting" is less than the proportion of boys. Therefore, the alternative hypothesis is:
$H_{1}: p_{B} > p_{G}$

STEP 4

STEP 5

Find the p-value using the z-test statistic obtained in STEP_2. Since this is a one-tailed test, find the probability that $Z$ is greater than the calculated z-value.

STEP 6

Make a decision. Compare the p-value to the significance level $\alpha = 0.01$ .
- If $p \leq 0.01$ , reject the null hypothesis. - If $p > 0.01$ , fail to reject the null hypothesis.

STEP 7

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Studdy Solution

STEP 1

1. We are conducting a hypothesis test for the difference in proportions between two independent groups (boys and girls).2. The level of significance is α=0.01 \alpha = 0.01 α=0.01.

STEP 2

1. Identify the correct alternative hypothesis.2. Calculate the test statistic.3. Find the p-value.4. Make a decision based on the p-value.5. Write the conclusion.

STEP 3

Identify the correct alternative hypothesis. The researcher believes that the proportion of girls involved in "sexting" is less than the proportion of boys. Therefore, the alternative hypothesis is:H1:pB>pG H_{1}: p_{B} > p_{G} H1​:pB​>pG​

STEP 4

STEP 5

Find the p-value using the z-test statistic obtained in STEP_2. Since this is a one-tailed test, find the probability that Z Z Z is greater than the calculated z-value.

STEP 6

Make a decision. Compare the p-value to the significance level α=0.01 \alpha = 0.01 α=0.01.- If p≤0.01 p \leq 0.01 p≤0.01, reject the null hypothesis. - If p>0.01 p > 0.01 p>0.01, fail to reject the null hypothesis.

STEP 7

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1. We are conducting a hypothesis test for the difference in proportions between two independent groups (boys and girls).
2. The level of significance is $\alpha = 0.01$ .

1. Identify the correct alternative hypothesis.
2. Calculate the test statistic.
3. Find the p-value.
4. Make a decision based on the p-value.
5. Write the conclusion.

Identify the correct alternative hypothesis. The researcher believes that the proportion of girls involved in "sexting" is less than the proportion of boys. Therefore, the alternative hypothesis is:
$H_{1}: p_{B} > p_{G}$

Find the p-value using the z-test statistic obtained in STEP_2. Since this is a one-tailed test, find the probability that $Z$ is greater than the calculated z-value.

Make a decision. Compare the p-value to the significance level $\alpha = 0.01$ .
- If $p \leq 0.01$ , reject the null hypothesis. - If $p > 0.01$ , fail to reject the null hypothesis.