Math / Data & Statistics

Question3. Psychologists were interested in the physiological changes that occur when a person laughed. The subjects in the study watched videos to encourage laughter. While the subjects were laughing, the researchers measured the subject's pulse. In a sample of 90 subjects, the mean heart rate was 73.5 beats per minute (bpm) with standard deviation 6 bpm . The mean resting heart rate of the population of all adults is known to be 71 bpm with standard deviation 7.5 bpm . The researchers wish to determine if the mean heart rate is elevated during laughter. In a previous example, we determined that the z -test for the mean should be conducted. a. Set up the null and alternative hypothesis for this test. b. Calculate the value of the test statistic. c. Find the p-value. d. At the $\alpha=0.05$ level of significance, make a decision whether to reject $\mathrm{H}_{0}$ . e. Can it be determined that the mean heart rate is elevated during laughter?

Studdy Solution

STEP 1

1. The sample size is $n = 90$ .
2. The sample mean heart rate during laughter is $\bar{x} = 73.5$ bpm.
3. The sample standard deviation is $s = 6$ bpm.
4. The population mean resting heart rate is $\mu_0 = 71$ bpm.
5. The population standard deviation is $\sigma = 7.5$ bpm.
6. We are conducting a one-sample z-test for the mean.
7. The level of significance is $\alpha = 0.05$ .

STEP 2

1. Set up the null and alternative hypotheses.
2. Calculate the test statistic.
3. Find the p-value.
4. Make a decision based on the p-value and significance level.
5. Interpret the results.

STEP 3

Set up the null and alternative hypotheses.
- Null hypothesis ( $H_0$ ): The mean heart rate during laughter is equal to the mean resting heart rate. $H_0: \mu = 71 \, \text{bpm}$
- Alternative hypothesis ( $H_a$ ): The mean heart rate during laughter is greater than the mean resting heart rate. $H_a: \mu > 71 \, \text{bpm}$

STEP 4

Calculate the test statistic using the formula for the z-test:
$z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}$
Substitute the given values:
$z = \frac{73.5 - 71}{7.5 / \sqrt{90}}$
Calculate the standard error:
$\text{Standard Error} = \frac{7.5}{\sqrt{90}} \approx 0.7906$
Calculate the z-value:
$z = \frac{2.5}{0.7906} \approx 3.16$

STEP 5

Find the p-value for the calculated z-value.
Since this is a one-tailed test, we look up the p-value for $z = 3.16$ in the standard normal distribution table.
The p-value is approximately:
$p \approx 0.0008$

STEP 6

Make a decision based on the p-value and the significance level $\alpha = 0.05$ .
Since $p = 0.0008 < 0.05$ , we reject the null hypothesis $H_0$ .

STEP 7

Interpret the results.
Since we rejected the null hypothesis, we have sufficient evidence to conclude that the mean heart rate is elevated during laughter compared to the resting heart rate.

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Studdy Solution

STEP 1

STEP 2

1. Set up the null and alternative hypotheses.
2. Calculate the test statistic.
3. Find the p-value.
4. Make a decision based on the p-value and significance level.
5. Interpret the results.

STEP 3

STEP 4

STEP 5

Find the p-value for the calculated z-value.
Since this is a one-tailed test, we look up the p-value for $z = 3.16$ in the standard normal distribution table.
The p-value is approximately:
$p \approx 0.0008$

STEP 6

Make a decision based on the p-value and the significance level $\alpha = 0.05$ .
Since $p = 0.0008 < 0.05$ , we reject the null hypothesis $H_0$ .

STEP 7

Interpret the results.
Since we rejected the null hypothesis, we have sufficient evidence to conclude that the mean heart rate is elevated during laughter compared to the resting heart rate.

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Studdy Solution

STEP 1

STEP 2

1. Set up the null and alternative hypotheses.2. Calculate the test statistic.3. Find the p-value.4. Make a decision based on the p-value and significance level.5. Interpret the results.

STEP 3

STEP 4

STEP 5

Find the p-value for the calculated z-value.Since this is a one-tailed test, we look up the p-value for z=3.16 z = 3.16 z=3.16 in the standard normal distribution table.The p-value is approximately:p≈0.0008p \approx 0.0008p≈0.0008

STEP 6

Make a decision based on the p-value and the significance level α=0.05 \alpha = 0.05 α=0.05.Since p=0.0008<0.05 p = 0.0008 < 0.05 p=0.0008<0.05, we reject the null hypothesis H0 H_0 H0​.

STEP 7

Interpret the results.Since we rejected the null hypothesis, we have sufficient evidence to conclude that the mean heart rate is elevated during laughter compared to the resting heart rate.

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

1. Set up the null and alternative hypotheses.
2. Calculate the test statistic.
3. Find the p-value.
4. Make a decision based on the p-value and significance level.
5. Interpret the results.

Find the p-value for the calculated z-value.
Since this is a one-tailed test, we look up the p-value for $z = 3.16$ in the standard normal distribution table.
The p-value is approximately:
$p \approx 0.0008$

Make a decision based on the p-value and the significance level $\alpha = 0.05$ .
Since $p = 0.0008 < 0.05$ , we reject the null hypothesis $H_0$ .

Interpret the results.
Since we rejected the null hypothesis, we have sufficient evidence to conclude that the mean heart rate is elevated during laughter compared to the resting heart rate.