Questions: A data set includes the counts of chocolate chips from three different types of Chips Ahoy cookies. The accompanying StatCrunch display shows results from analysis of variance used with those three types of cookies. Use a 0.05 significance level to test the claim that the three different types of cookies have the same mean number of chocolate chips. ANOVA table: - Source DF SS MS F-Stat P-value - Columns 2 1134.0000 567.00000 57.6382 0.0001 - Error 77 757.46667 9.8372295 - Total 79 1891.4667 Determine the null hypothesis H0 Determine the alternative hypothesis. H1 Determine the test statistic The test statistic is (Round to two decimal places as needed.) Determine the P-value. The P-value is

A data set includes the counts of chocolate chips from three different types of Chips Ahoy cookies. The accompanying StatCrunch display shows results from analysis of variance used with those three types of cookies. Use a 0.05 significance level to test the claim that the three different types of cookies have the same mean number of chocolate chips. ANOVA table: - Source DF SS MS F-Stat P-value - Columns 2 1134.0000 567.00000 57.6382 0.0001 - Error 77 757.46667 9.8372295 - Total 79 1891.4667 Determine the null hypothesis H0 Determine the alternative hypothesis. H1 Determine the test statistic The test statistic is (Round to two decimal places as needed.) Determine the P-value. The P-value is
Transcript text: A data set includes the counts of chocolate chips from three different types of Chips Ahoy cookies. The accompanying StatCrunch display shows results from analysis of variance used with those three types of cookies. Use a 0.05 significance level to test the claim that the three different types of cookies have the same mean number of chocolate chips \begin{tabular}{|l|r|c|c|c|c|} \hline \multicolumn{6}{|l|}{ ANOVA table } \\ \hline \multicolumn{1}{|c|}{ Source } & \multicolumn{1}{l|}{ DF } & SS & MS & F-Stat & P-value \\ \hline Columns & 2 & 1134.0000 & 567.00000 & 57.6382 & 0.0001 \\ \hline Error & 77 & 757.46667 & 9.8372295 & & \\ \hline Total & 79 & 1891.4667 & & & \\ \hline \end{tabular} Determine the null hypothesis $\mathrm{H}_{0}$ $\square$ Determine the alternative hypothesis. $\mathrm{H}_{1}$ $\square$ Determine the test statistic The test statistic is $\square$ (Round to two decimal places as needed.) Determine the $P$-value. The P-value is $\square$ $\square$
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Solution

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Solution Steps

To solve this problem, we need to perform an analysis of variance (ANOVA) test to determine if there is a significant difference in the mean number of chocolate chips among the three types of cookies. Here are the steps:

  1. Determine the null hypothesis (H0): The null hypothesis states that the mean number of chocolate chips is the same for all three types of cookies.
  2. Determine the alternative hypothesis (H1): The alternative hypothesis states that at least one type of cookie has a different mean number of chocolate chips.
  3. Determine the test statistic: The test statistic (F-statistic) is given in the ANOVA table.
  4. Determine the P-value: The P-value is also provided in the ANOVA table.
Step 1: Null Hypothesis

The null hypothesis (\( \mathrm{H}_{0} \)) states that the mean number of chocolate chips is the same for all three types of cookies. Mathematically, this can be expressed as: \[ \mathrm{H}_{0}: \mu_1 = \mu_2 = \mu_3 \]

Step 2: Alternative Hypothesis

The alternative hypothesis (\( \mathrm{H}_{1} \)) states that at least one type of cookie has a different mean number of chocolate chips. This can be expressed as: \[ \mathrm{H}_{1}: \text{At least one } \mu_i \text{ is different} \]

Step 3: Test Statistic

The test statistic (F-statistic) calculated from the ANOVA table is: \[ F = 57.64 \]

Step 4: P-value

The P-value associated with the test statistic is: \[ P = 0.0001 \]

Final Answer

  • Null Hypothesis: \( \mathrm{H}_{0}: \mu_1 = \mu_2 = \mu_3 \)
  • Alternative Hypothesis: \( \mathrm{H}_{1}: \text{At least one } \mu_i \text{ is different} \)
  • Test Statistic: \( F = 57.64 \)
  • P-value: \( P = 0.0001 \)

Thus, the final boxed answers are: \[ \boxed{\mathrm{H}_{0}: \mu_1 = \mu_2 = \mu_3} \] \[ \boxed{\mathrm{H}_{1}: \text{At least one } \mu_i \text{ is different}} \] \[ \boxed{F = 57.64} \] \[ \boxed{P = 0.0001} \]

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