Questions: At the α=0.05 level of significance, test to see whether the mean drying time is the same. State the null and alternative hypotheses. H0: μ1=μ2=μ3=μ4 Ha: Not all the population means are equal. Find the value of the test statistic. (Round your answer to two decimal places.) 2.51 Find the p-value. (Round your answer to four decimal places.) p-value =0.0970 What is your conclusion? Do not reject H0. There is not sufficient evidence to conclude that the mean drying times

At the α=0.05 level of significance, test to see whether the mean drying time is the same. State the null and alternative hypotheses.
H0: μ1=μ2=μ3=μ4
Ha: Not all the population means are equal.

Find the value of the test statistic. (Round your answer to two decimal places.)
2.51

Find the p-value. (Round your answer to four decimal places.)
p-value =0.0970

What is your conclusion?
Do not reject H0. There is not sufficient evidence to conclude that the mean drying times

Transcript text: At the $\alpha=0.05$ level of significance, test to see whether the mean drying time is the same. State the null and alternative hypotheses. $H_{0}: \mu_{1}=\mu_{2}=\mu_{3}=\mu_{4}$ $H_{\mathrm{a}}$ : Not all the population means are equal. Find the value of the test statistic. (Round your answer to two decimal places.) 2.51 Find the $p$-value. (Round your answer to four decimal places.) $p$-value $=0.0970$ What is your conclusion? Do not reject $H_{0}$. There is not sufficient evidence to conclude that the mean drying times

Solution

Solution Steps

To solve this problem, we need to perform an ANOVA (Analysis of Variance) test to determine if there are any statistically significant differences between the means of the groups. The null hypothesis states that all group means are equal, while the alternative hypothesis states that at least one group mean is different. We will calculate the test statistic and the p-value to make a decision about the null hypothesis.

Step 1: State the Null and Alternative Hypotheses

The problem provides several options for the null and alternative hypotheses. We need to select the correct pair for testing whether the mean drying times are the same.

Null Hypothesis ($H_0$): The mean drying times for all groups are equal, i.e., $ \mu_1 = \mu_2 = \mu_3 = \mu_4 $.
Alternative Hypothesis ($H_a$): Not all the population means are equal.

Thus, the correct hypotheses are:

\[ \begin{array}{c} H_{0}: \mu_{1}=\mu_{2}=\mu_{3}=\mu_{4} \\ H_{a}: \text{Not all the population means are equal.} \end{array} \]

Step 2: Find the Value of the Test Statistic

The problem states that the test statistic value is given as 2.51. This value is typically calculated using an ANOVA test when comparing more than two means.

Step 3: Find the $p$-value

The problem provides the $p$-value as 0.0970. This value is used to determine the statistical significance of the test statistic.

Step 4: Conclusion Based on the $p$-value

To make a conclusion, we compare the $p$-value to the significance level $\alpha = 0.05$.

If $p$-value $\leq \alpha$, we reject the null hypothesis.
If $p$-value $> \alpha$, we do not reject the null hypothesis.

Given that the $p$-value is 0.0970, which is greater than 0.05, we do not reject the null hypothesis.

Final Answer

Hypotheses: \[ \begin{array}{c} H_{0}: \mu_{1}=\mu_{2}=\mu_{3}=\mu_{4} \\ H_{a}: \text{Not all the population means are equal.} \end{array} \]
Test Statistic: $\boxed{2.51}$
$p$-value: $\boxed{0.0970}$
Conclusion: Do not reject $H_0$. There is not sufficient evidence to conclude that the mean drying times for the groups are different.

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