Math / Data & Statistics

QuestionALL 2 2024) Kerlise Sylvestre nference Homework Question 4, 9.1.11-T HW Score: 10.94\%, 2.63 of 24 points Part 5 of 7 Points: 0 of 1 Save
A study was conducted to determine the proportion of people who dream in black and white instead of color. Among 315 people over the age of 55,68 dream in black and white, and among 292 people under the age of 25,13 dream in black and white. Use a 0.01 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. $\mathrm{H}_{1}: \mathrm{P}_{1}<\mathrm{P}_{2}$ $H_{1}: p_{1}>p_{2}$ $H_{1}: P_{1} \mp P_{2}$
Identify the test statistic. $z=6.21$ (Round to two decimal places as needed.) Identify the P -value. $P \text {-value }=0.000$ (Round to three decimal places as needed.) What is the conclusion based on the hypothesis test? The P -value is less than $\square$ reject the null hypothesis. There is $\square$ evidence to support the claim that the proportion of people over 55 who dream in black and white is greater than the proportion for those under 25. b. Test the claim by constructing an appropriate confidence interval. $\square$ The $98 \%$ confidence interval is $\square<\left(p_{1}-p_{2}\right)<\square$ . (Round to three decimal places as needed.) Clear all Check answer $12: 47$ PM 11/22/2024

Studdy Solution

STEP 1

What is this asking? We want to see if older people (over 55) dream in black and white more often than younger people (under 25), and we're using a special statistical test to do it! Watch out! Don't mix up the groups (over 55 vs. under 25) or the proportions!
Also, make sure to understand what the p-value really means.

STEP 2

1. State the Hypotheses
2. Calculate the Pooled Proportion
3. Calculate the Test Statistic
4. Find the P-value
5. Construct the Confidence Interval
6. State the Conclusion

STEP 3

Let $p_1$ be the proportion of people over 55 who dream in black and white, and $p_2$ be the proportion of people under 25 who dream in black and white.

STEP 4

Our null hypothesis ( $H_0$ ) is that there's no difference between the proportions: $p_1 = p_2$ .

STEP 5

Our alternative hypothesis ( $H_1$ ) is that the proportion for older people is greater than the proportion for younger people: $p_1 > p_2$ .

STEP 6

The pooled proportion ( $\hat{p}$ ) is the combined proportion of both groups who dream in black and white.
It's like mixing all the dreamers together!

STEP 7

We have 68 black and white dreamers out of 315 older people, and 13 black and white dreamers out of 292 younger people.
So, the total number of black and white dreamers is $68 + 13 = \mathbf{81}$ , and the total number of people is $315 + 292 = \mathbf{607}$ .

STEP 8

The pooled proportion is calculated as: $\hat{p} = \frac{\text{Total black and white dreamers}}{\text{Total people}} = \frac{81}{607} \approx \mathbf{0.133}$

STEP 9

The test statistic (z) tells us how far apart the two proportions are, in terms of standard deviations.

STEP 10

The formula for the test statistic is: $z = \frac{(\hat{p}_1 - \hat{p}_2) - 0}{\sqrt{\hat{p}(1 - \hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}}$ where $\hat{p}_1 = \frac{68}{315} \approx 0.216$ , $\hat{p}_2 = \frac{13}{292} \approx 0.045$ , $n_1 = 315$ , and $n_2 = 292$ .

STEP 11

Plugging in the values, we get: $z = \frac{(0.216 - 0.045) - 0}{\sqrt{0.133(1 - 0.133)(\frac{1}{315} + \frac{1}{292})}} \approx \frac{0.171}{\sqrt{0.115 \cdot 0.0066}} \approx \frac{0.171}{0.0278} \approx \mathbf{6.15}$

STEP 12

The $p$ -value is the probability of getting a test statistic as extreme as ours (or more extreme) if the null hypothesis is true.
A tiny $p$ -value means it's unlikely that the null hypothesis is true!

STEP 13

Since our $z$ is approximately 6.15, and our alternative hypothesis is $p_1 > p_2$ (a one-tailed test), our $p$ -value is extremely small, close to 0.000.

STEP 14

A 98% confidence interval gives us a range of plausible values for the difference between the two proportions ( $p_1 - p_2$ ).

STEP 15

The formula is: $(\hat{p}_1 - \hat{p}_2) \pm z_{\alpha/2} \cdot \sqrt{\frac{\hat{p}_1(1 - \hat{p}_1)}{n_1} + \frac{\hat{p}_2(1 - \hat{p}_2)}{n_2}}$ where $z_{\alpha/2} = 2.33$ for a 98% confidence level.

STEP 16

Plugging in the values, we get: $(0.216 - 0.045) \pm 2.33 \cdot \sqrt{\frac{0.216(1 - 0.216)}{315} + \frac{0.045(1 - 0.045)}{292}}$ $0.171 \pm 2.33 \cdot \sqrt{0.00054 + 0.00015} \approx 0.171 \pm 2.33 \cdot 0.026 \approx 0.171 \pm 0.061$ So, the interval is approximately $(0.110, 0.232)$ .

STEP 17

Our $p$ -value is close to zero, which is less than our significance level of 0.01.
This means we reject the null hypothesis!

STEP 18

Our 98% confidence interval for $(p_1 - p_2)$ is $(0.110, 0.232)$ , which does not contain zero.
This further supports rejecting the null hypothesis.

STEP 19

The test statistic is approximately 6.15.
The $p$ -value is approximately 0.000.
We reject the null hypothesis.
There is strong evidence to support the claim that the proportion of people over 55 who dream in black and white is greater than the proportion for those under 25.
The 98% confidence interval for the difference in proportions is (0.110, 0.232).

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Studdy Solution

STEP 1

STEP 2

1. State the Hypotheses2. Calculate the Pooled Proportion3. Calculate the Test Statistic4. Find the P-value5. Construct the Confidence Interval6. State the Conclusion

STEP 3

Let p1p_1p1​ be the proportion of people over 55 who dream in black and white, and p2p_2p2​ be the proportion of people under 25 who dream in black and white.

STEP 4

Our **null hypothesis** (H0H_0H0​) is that there's *no difference* between the proportions: p1=p2p_1 = p_2p1​=p2​.

STEP 5

Our **alternative hypothesis** (H1H_1H1​) is that the proportion for older people is *greater* than the proportion for younger people: p1>p2p_1 > p_2p1​>p2​.

STEP 6

The **pooled proportion** (p^\hat{p}p^​) is the combined proportion of *both* groups who dream in black and white.It's like mixing all the dreamers together!

STEP 7

We have 68 black and white dreamers out of 315 older people, and 13 black and white dreamers out of 292 younger people.So, the **total number of black and white dreamers** is 68+13=8168 + 13 = \mathbf{81}68+13=81, and the **total number of people** is 315+292=607315 + 292 = \mathbf{607}315+292=607.

STEP 8

The **pooled proportion** is calculated as: p^=Total black and white dreamersTotal people=81607≈0.133\hat{p} = \frac{\text{Total black and white dreamers}}{\text{Total people}} = \frac{81}{607} \approx \mathbf{0.133}p^​=Total peopleTotal black and white dreamers​=60781​≈0.133

STEP 9

The **test statistic** (z) tells us how far apart the two proportions are, in terms of standard deviations.

STEP 10

STEP 11

STEP 12

The **ppp-value** is the probability of getting a test statistic as extreme as ours (or more extreme) if the null hypothesis is true.A tiny ppp-value means it's *unlikely* that the null hypothesis is true!

STEP 13

Since our zzz is approximately 6.15, and our alternative hypothesis is p1>p2p_1 > p_2p1​>p2​ (a one-tailed test), our ppp-value is extremely small, close to **0.000**.

STEP 14

A 98% confidence interval gives us a range of plausible values for the *difference* between the two proportions (p1−p2p_1 - p_2p1​−p2​).

STEP 15

STEP 16

STEP 17

Our **ppp-value** is close to zero, which is *less than* our significance level of 0.01.This means we **reject the null hypothesis**!

STEP 18

Our 98% confidence interval for (p1−p2)(p_1 - p_2)(p1​−p2​) is (0.110,0.232)(0.110, 0.232)(0.110,0.232), which does *not* contain zero.This further supports rejecting the null hypothesis.

STEP 19

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1. State the Hypotheses
2. Calculate the Pooled Proportion
3. Calculate the Test Statistic
4. Find the P-value
5. Construct the Confidence Interval
6. State the Conclusion

Let $p_1$ be the proportion of people over 55 who dream in black and white, and $p_2$ be the proportion of people under 25 who dream in black and white.

Our null hypothesis ( $H_0$ ) is that there's no difference between the proportions: $p_1 = p_2$ .

Our alternative hypothesis ( $H_1$ ) is that the proportion for older people is greater than the proportion for younger people: $p_1 > p_2$ .

The pooled proportion ( $\hat{p}$ ) is the combined proportion of both groups who dream in black and white.
It's like mixing all the dreamers together!

We have 68 black and white dreamers out of 315 older people, and 13 black and white dreamers out of 292 younger people.
So, the total number of black and white dreamers is $68 + 13 = \mathbf{81}$ , and the total number of people is $315 + 292 = \mathbf{607}$ .

The pooled proportion is calculated as: $\hat{p} = \frac{\text{Total black and white dreamers}}{\text{Total people}} = \frac{81}{607} \approx \mathbf{0.133}$

The test statistic (z) tells us how far apart the two proportions are, in terms of standard deviations.

The $p$ -value is the probability of getting a test statistic as extreme as ours (or more extreme) if the null hypothesis is true.
A tiny $p$ -value means it's unlikely that the null hypothesis is true!

Since our $z$ is approximately 6.15, and our alternative hypothesis is $p_1 > p_2$ (a one-tailed test), our $p$ -value is extremely small, close to 0.000.

A 98% confidence interval gives us a range of plausible values for the difference between the two proportions ( $p_1 - p_2$ ).

Our $p$ -value is close to zero, which is less than our significance level of 0.01.
This means we reject the null hypothesis!

Our 98% confidence interval for $(p_1 - p_2)$ is $(0.110, 0.232)$ , which does not contain zero.
This further supports rejecting the null hypothesis.