Math / Data & Statistics

QuestionA study was conducted to determine the proportion of people who dream in black and white instead of color. Among 323 people over the age of 55,72 dream in black and white, and among 289 people under the age of 25,16 dream in black and white. Use a 0.05 significance level to test the claim that the proportion of people over 55 who dream in black and white is greater than the proportion for those under 25. Complete parts (a) through (c) below. a. Test the claim using a hypothesis test.
Consider the first sample to be the sample of people over the age of 55 and the second sample to be the sample of people under the age of 25 . What are the null and alternative hypotheses for the hypothesis test? A. $H_{0}: P_{1}=P_{2}$ B. $H_{0}: p_{1}=p_{2}$ C. $H_{0}: p_{1} \leq p_{2}$ $H_{1}: p_{1} \neq p_{2}$ $H_{1}: p_{1}>p_{2}$ $H_{1}: p_{1} \neq p_{2}$ D. $H_{0}: p_{1}=p_{2}$ E. $H_{0}: p_{1} \geq p_{2}$ $H_{1}: P_{1}<P_{2}$ $\mathrm{H}_{1}: \mathrm{p}_{1} \neq \mathrm{p}_{2}$ F. $H_{0}: p_{1} \neq p_{2}$ $H_{1}: p_{1}=p_{2}$
Identify the test statistic. $\mathrm{z}=\square$ (Round to two decimal places as needed.)

Studdy Solution

STEP 1

1. We are comparing two independent proportions.
2. The significance level is $\alpha = 0.05$ .
3. The sample sizes are sufficiently large to use a normal approximation.

STEP 2

1. State the null and alternative hypotheses.
2. Calculate the sample proportions.
3. Compute the pooled sample proportion.
4. Calculate the test statistic.
5. Make a decision based on the test statistic and significance level.

STEP 3

State the null and alternative hypotheses. Since we are testing if the proportion of people over 55 who dream in black and white is greater than the proportion for those under 25, the hypotheses are:
- Null hypothesis ( $H_0$ ): $p_1 = p_2$ - Alternative hypothesis ( $H_1$ ): $p_1 > p_2$
The correct choice is:
$\text{C. } H_0: p_1 = p_2, \quad H_1: p_1 > p_2$

STEP 4

Calculate the sample proportions:
- For people over 55: $\hat{p}_1 = \frac{72}{323}$ - For people under 25: $\hat{p}_2 = \frac{16}{289}$

STEP 5

Compute the pooled sample proportion:
$\hat{p} = \frac{72 + 16}{323 + 289} = \frac{88}{612}$

STEP 6

Calculate the test statistic using the formula for the difference in proportions:
$z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1 - \hat{p})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}}$
where $n_1 = 323$ and $n_2 = 289$ .
Substitute the values:
$z = \frac{\frac{72}{323} - \frac{16}{289}}{\sqrt{\frac{88}{612} \left(1 - \frac{88}{612}\right) \left(\frac{1}{323} + \frac{1}{289}\right)}}$
Calculate the numerical value of $z$ and round to two decimal places.

STEP 7

Make a decision based on the test statistic and the significance level $\alpha = 0.05$ .
- If $z$ is greater than the critical value from the standard normal distribution for a one-tailed test at $\alpha = 0.05$ , reject $H_0$ . - Otherwise, do not reject $H_0$ .
The test statistic $z$ is approximately:
$z = \boxed{3.15}$

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Studdy Solution

STEP 1

1. We are comparing two independent proportions.
2. The significance level is $\alpha = 0.05$ .
3. The sample sizes are sufficiently large to use a normal approximation.

STEP 2

1. State the null and alternative hypotheses.
2. Calculate the sample proportions.
3. Compute the pooled sample proportion.
4. Calculate the test statistic.
5. Make a decision based on the test statistic and significance level.

STEP 3

STEP 4

Calculate the sample proportions:
- For people over 55: $\hat{p}_1 = \frac{72}{323}$ - For people under 25: $\hat{p}_2 = \frac{16}{289}$

STEP 5

Compute the pooled sample proportion:
$\hat{p} = \frac{72 + 16}{323 + 289} = \frac{88}{612}$

STEP 6

STEP 7

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Studdy Solution

STEP 1

1. We are comparing two independent proportions.2. The significance level is α=0.05 \alpha = 0.05 α=0.05.3. The sample sizes are sufficiently large to use a normal approximation.

STEP 2

1. State the null and alternative hypotheses.2. Calculate the sample proportions.3. Compute the pooled sample proportion.4. Calculate the test statistic.5. Make a decision based on the test statistic and significance level.

STEP 3

STEP 4

Calculate the sample proportions:- For people over 55: p^1=72323 \hat{p}_1 = \frac{72}{323} p^​1​=32372​ - For people under 25: p^2=16289 \hat{p}_2 = \frac{16}{289} p^​2​=28916​

STEP 5

Compute the pooled sample proportion:p^=72+16323+289=88612 \hat{p} = \frac{72 + 16}{323 + 289} = \frac{88}{612} p^​=323+28972+16​=61288​

STEP 6

STEP 7

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1. We are comparing two independent proportions.
2. The significance level is $\alpha = 0.05$ .
3. The sample sizes are sufficiently large to use a normal approximation.

1. State the null and alternative hypotheses.
2. Calculate the sample proportions.
3. Compute the pooled sample proportion.
4. Calculate the test statistic.
5. Make a decision based on the test statistic and significance level.

Calculate the sample proportions:
- For people over 55: $\hat{p}_1 = \frac{72}{323}$ - For people under 25: $\hat{p}_2 = \frac{16}{289}$

Compute the pooled sample proportion:
$\hat{p} = \frac{72 + 16}{323 + 289} = \frac{88}{612}$