Math / Data & Statistics

QuestionThe average salary for American college graduates is $\$ 48,400$ . You suspect that the average is more for graduates from your college. The 53 randomly selected graduates from your college had an average salary of $\$ 51,937$ and a standard deviation of $\$ 9,140$ . What can be concluded at the $\alpha=0.05$ level of significance? a. For this study, we should use Select an answer b. The null and alternative hypotheses would be: $H_{0}$ : ? 웅 Select an answer 0 $\square$ $H_{1}$ : ? . 0 Select an answer $\square$ c. The test statistic ? ${ }^{0}=$ $\square$ (please show your answer to 4 decimal places.) d. The $p$ -value $=$ $\square$ (Please show your answer to 4 decimal places.) e. The $p$ -value is $\square$ $\alpha$ f. Based on this, we should Select an answer the null hypothesis. g. Thus, the final conclusion is that ... The data suggest that the sample mean is not significantly greater than 48,400 at $\alpha=0.05$ , so there is statistically insignificant evidence to conclude that the sample mean salary for graduates from your college is greater than 51,937 . The data suggest that the population mean is not significantly greater than 48,400 at $\alpha=$ 0.05 , so there is statistically insignificant evidence to conclude that the population mean salary for graduates from your college is greater than 48,400. The data suggest that the populaton mean is significantly greater than 48,400 at $\alpha=0.05$ , so there is statistically significant evidence to conclude that the population mean salary for graduates from your college is greater than 48,400 . h. Interpret the $p$ -value in the context of the study. There is a $0.34187113 \%$ chance of a Type I error. If the population mean salary for graduates from your college is $\$ 48,400$ and if another 53 graduates from your college are surveyed then there would be a $0.34187113 \%$ chance that the population mean salary for graduates from your college would be greater than $\$ 48,400$ . If the population mean salary for graduates from your college is $\$ 48,400$ and if another 53 graduates from your college are surveyed then there would be a $0.34187113 \%$ chance that the sample mean for these 53 graduates from your college surveyed would be greater than \ $51,937. There is a$ 0.34187113 \% $chance that the population mean salary for graduates from your college is greater than$ \ $48,400$ .

Studdy Solution

STEP 1

1. The sample size is 53 graduates.
2. The sample mean salary is \$51,937.
3. The population mean salary for American college graduates is \$48,400.
4. The sample standard deviation is \$9,140.
5. The significance level $\alpha$ is 0.05.
6. We are conducting a one-sample t-test to determine if the sample mean is significantly greater than the population mean.

STEP 2

1. Identify the appropriate statistical test.
2. State the null and alternative hypotheses.
3. Calculate the test statistic.
4. Determine the p-value.
5. Compare the p-value to $\alpha$ .
6. Make a decision regarding the null hypothesis.
7. Draw a conclusion from the test.
8. Interpret the p-value in context.

STEP 3

For this study, we should use a one-sample t-test because we are comparing the sample mean to a known population mean with an unknown population standard deviation.

STEP 4

State the null and alternative hypotheses:
$H_0: \mu = 48,400$ $H_1: \mu > 48,400$

STEP 5

Calculate the test statistic using the formula:
$t = \frac{\bar{x} - \mu}{s / \sqrt{n}}$
Where: $\bar{x} = 51,937$ , $\mu = 48,400$ , $s = 9,140$ , $n = 53$ .
$t = \frac{51,937 - 48,400}{9,140 / \sqrt{53}}$
$t \approx 2.7489$

STEP 6

Determine the p-value for the calculated t-statistic with $n-1 = 52$ degrees of freedom.
Using a t-distribution table or calculator, find the p-value:
$p \approx 0.0034$

STEP 7

Compare the p-value to $\alpha$ :
$p = 0.0034$ $\alpha = 0.05$
Since $p < \alpha$ , we reject the null hypothesis.

STEP 8

Based on this, we should reject the null hypothesis.

STEP 9

Thus, the final conclusion is that the data suggest that the population mean is significantly greater than \$48,400 at $\alpha = 0.05$, so there is statistically significant evidence to conclude that the population mean salary for graduates from your college is greater than \$48,400.

STEP 10

Interpret the p-value in the context of the study:
If the population mean salary for graduates from your college is \$48,400 and if another 53 graduates from your college are surveyed, then there would be a 0.34% chance that the sample mean for these 53 graduates from your college surveyed would be greater than \$51,937.

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Studdy Solution

STEP 1

STEP 2

STEP 3

For this study, we should use a one-sample t-test because we are comparing the sample mean to a known population mean with an unknown population standard deviation.

STEP 4

State the null and alternative hypotheses:
$H_0: \mu = 48,400$ $H_1: \mu > 48,400$

STEP 5

Calculate the test statistic using the formula:
$t = \frac{\bar{x} - \mu}{s / \sqrt{n}}$
Where: $\bar{x} = 51,937$ , $\mu = 48,400$ , $s = 9,140$ , $n = 53$ .
$t = \frac{51,937 - 48,400}{9,140 / \sqrt{53}}$
$t \approx 2.7489$

STEP 6

Determine the p-value for the calculated t-statistic with $n-1 = 52$ degrees of freedom.
Using a t-distribution table or calculator, find the p-value:
$p \approx 0.0034$

STEP 7

Compare the p-value to $\alpha$ :
$p = 0.0034$ $\alpha = 0.05$
Since $p < \alpha$ , we reject the null hypothesis.

STEP 8

Based on this, we should reject the null hypothesis.

STEP 9

Thus, the final conclusion is that the data suggest that the population mean is significantly greater than \$48,400 at \(\alpha = 0.05\), so there is statistically significant evidence to conclude that the population mean salary for graduates from your college is greater than \$48,400.

STEP 10

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Studdy Solution

STEP 1

STEP 2

STEP 3

For this study, we should use a one-sample t-test because we are comparing the sample mean to a known population mean with an unknown population standard deviation.

STEP 4

State the null and alternative hypotheses:H0:μ=48,400 H_0: \mu = 48,400 H0​:μ=48,400 H1:μ>48,400 H_1: \mu > 48,400 H1​:μ>48,400

STEP 5

STEP 6

Determine the p-value for the calculated t-statistic with n−1=52n-1 = 52n−1=52 degrees of freedom.Using a t-distribution table or calculator, find the p-value:p≈0.0034 p \approx 0.0034 p≈0.0034

STEP 7

Compare the p-value to α\alphaα:p=0.0034 p = 0.0034 p=0.0034 α=0.05 \alpha = 0.05 α=0.05Since p<α p < \alpha p<α, we reject the null hypothesis.

STEP 8

Based on this, we should reject the null hypothesis.

STEP 9

Thus, the final conclusion is that the data suggest that the population mean is significantly greater than \$48,400 at \(\alpha = 0.05\), so there is statistically significant evidence to conclude that the population mean salary for graduates from your college is greater than \$48,400.

STEP 10

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

State the null and alternative hypotheses:
$H_0: \mu = 48,400$ $H_1: \mu > 48,400$

Determine the p-value for the calculated t-statistic with $n-1 = 52$ degrees of freedom.
Using a t-distribution table or calculator, find the p-value:
$p \approx 0.0034$

Compare the p-value to $\alpha$ :
$p = 0.0034$ $\alpha = 0.05$
Since $p < \alpha$ , we reject the null hypothesis.