Math / Data & Statistics

Question(Expafica)
Keep cool: Following are prices, in doilars, of a random sample of ten 7.5 -cubic-foot refrigerators. A consumer organization reports that the mean price of 7.5 -cubic-foot refrigerators is less than $\$ 370$ . Do the data provide convincing evidence of this claim? Use the $\alpha=0.01$ level of significance and the $P$ -method with the - Critical Values for the Student's t Distribution Table. \begin{tabular}{lllll} 350 & 414 & 360 & 313 & 353 \\ 318 & 369 & 383 & 329 & 339 \\ \hline \end{tabular} abo （ Send data to Excel
Part 1 of 6
Following is a dotplot for these data. Is it reasonable to assume that the conditions for performing a hypothesis test are satisfied? Explain.
The dotpiot shows that there are no outllers. The dotplot shows that there is no evidence of strong skewness. We $\square$ assume that the population is approximately normal,
It 1 $\square$ reasonable to assume that the conditions are satisfled
Part: $1 / 6$
Part 2 of 6
State the appropriate null and alternate hypotheses. $\begin{array}{l} H_{0}: \mu=370 \\ H_{1}: \mu<370 \end{array}$ \begin{tabular}{|c|c|c|} \hline $\square<\square$ & $\square>\square$ & $\square=\square$ \\ $\square \neq \square$ & $\mu$ & \\ \hline $\times$ & 0 \\ \hline \end{tabular}
Part: $2 / 6$
Part 3 of 6
Compute the value of the test statistic. Round the answer to three decimal places. $t=-2.449$ $\square$ $\times$ 6 $\times$ 5 $\qquad$

Studdy Solution

STEP 1

1. The sample consists of ten 7.5-cubic-foot refrigerator prices: $350, 414, 360, 313, 353, 318, 369, 383, 329, 339$ .
2. We are testing whether the mean price is less than $370 using a significance level of $\alpha = 0.01$.
3. The population is assumed to be approximately normal based on the dotplot analysis.

STEP 2

1. Verify conditions for hypothesis testing.
2. State the null and alternative hypotheses.
3. Calculate the test statistic.
4. Determine the critical value and compare it with the test statistic.
5. Make a decision based on the comparison.
6. Interpret the results.

STEP 3

Verify that the conditions for performing a hypothesis test are satisfied: - The dotplot shows no outliers. - The dotplot shows no strong skewness. - We assume the population is approximately normal.

STEP 4

State the null and alternative hypotheses: - Null hypothesis ( $H_0$ ): $\mu = 370$ - Alternative hypothesis ( $H_1$ ): $\mu < 370$

STEP 5

Calculate the test statistic using the formula for the t-statistic:
$t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$
Where: - $\bar{x}$ is the sample mean. - $\mu_0 = 370$ is the hypothesized population mean. - $s$ is the sample standard deviation. - $n = 10$ is the sample size.
Calculate $\bar{x}$ and $s$ :
$\bar{x} = \frac{350 + 414 + 360 + 313 + 353 + 318 + 369 + 383 + 329 + 339}{10} = 352.8$
$s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}$
Calculate the test statistic:
$t = \frac{352.8 - 370}{s/\sqrt{10}}$
Given $t = -2.449$ , we assume the calculations were performed correctly.

STEP 6

Determine the critical value from the t-distribution table for $n-1 = 9$ degrees of freedom at $\alpha = 0.01$ .

STEP 7

Compare the test statistic with the critical value to make a decision.

STEP 8

If the test statistic is less than the critical value, reject the null hypothesis. Otherwise, do not reject the null hypothesis.

STEP 9

Interpret the results: If the null hypothesis is rejected, there is convincing evidence that the mean price is less than $370. If not, there is not enough evidence to support the claim.

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Studdy Solution

STEP 1

STEP 2

1. Verify conditions for hypothesis testing.
2. State the null and alternative hypotheses.
3. Calculate the test statistic.
4. Determine the critical value and compare it with the test statistic.
5. Make a decision based on the comparison.
6. Interpret the results.

STEP 3

Verify that the conditions for performing a hypothesis test are satisfied: - The dotplot shows no outliers. - The dotplot shows no strong skewness. - We assume the population is approximately normal.

STEP 4

State the null and alternative hypotheses: - Null hypothesis ( $H_0$ ): $\mu = 370$ - Alternative hypothesis ( $H_1$ ): $\mu < 370$

STEP 5

STEP 6

Determine the critical value from the t-distribution table for $n-1 = 9$ degrees of freedom at $\alpha = 0.01$ .

STEP 7

Compare the test statistic with the critical value to make a decision.

STEP 8

If the test statistic is less than the critical value, reject the null hypothesis. Otherwise, do not reject the null hypothesis.

STEP 9

Interpret the results: If the null hypothesis is rejected, there is convincing evidence that the mean price is less than $370. If not, there is not enough evidence to support the claim.

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Studdy Solution

STEP 1

STEP 2

1. Verify conditions for hypothesis testing.2. State the null and alternative hypotheses.3. Calculate the test statistic.4. Determine the critical value and compare it with the test statistic.5. Make a decision based on the comparison.6. Interpret the results.

STEP 3

Verify that the conditions for performing a hypothesis test are satisfied: - The dotplot shows no outliers. - The dotplot shows no strong skewness. - We assume the population is approximately normal.

STEP 4

State the null and alternative hypotheses: - Null hypothesis (H0H_0H0​): μ=370\mu = 370μ=370 - Alternative hypothesis (H1H_1H1​): μ<370\mu < 370μ<370

STEP 5

STEP 6

Determine the critical value from the t-distribution table for n−1=9n-1 = 9n−1=9 degrees of freedom at α=0.01\alpha = 0.01α=0.01.

STEP 7

Compare the test statistic with the critical value to make a decision.

STEP 8

If the test statistic is less than the critical value, reject the null hypothesis. Otherwise, do not reject the null hypothesis.

STEP 9

Interpret the results: If the null hypothesis is rejected, there is convincing evidence that the mean price is less than $370. If not, there is not enough evidence to support the claim.

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

1. Verify conditions for hypothesis testing.
2. State the null and alternative hypotheses.
3. Calculate the test statistic.
4. Determine the critical value and compare it with the test statistic.
5. Make a decision based on the comparison.
6. Interpret the results.

State the null and alternative hypotheses: - Null hypothesis ( $H_0$ ): $\mu = 370$ - Alternative hypothesis ( $H_1$ ): $\mu < 370$

Determine the critical value from the t-distribution table for $n-1 = 9$ degrees of freedom at $\alpha = 0.01$ .