A story spoiler gives away the ending early. Does having a story spoiled in this way diminish suspense and hurt enjoyment? A study investigated this question. For 12 different short stories, the study's authors created a second versions in which a spoller paragraph at the beginning discusses the story and reveal the outcome. Each vers on of the 12 stories was read by at least 30 people and rated on a 1 to 10 scale to create an overall rating for the story (higher ratings indicate greater enjoyment). The results of the analysis are given below:
Two sample $\mathrm{T}$ hypothesis test:
$\mu_{1}:$ Mean of With spoiler
$\mu_{2}:$ Mean of Original
$\mu_{1}-\mu_{2}:$ Difference between two means
\[
\begin{array}{l}
H_{0}: \mu_{1}-\mu_{2}=0 \\
H_{A}: \mu_{1}-\mu_{2}< 0
\end{array}
\]
(without pooled variances)
Hypothesis test results:
\begin{tabular}{|l|r|c|c|c|c|}
\hline Difference & Sample Diff. & Std. Err. & DF & T-Stat & P-value \\
\hline$\mu_{1}-\mu_{2}$ & 0.51666667 & 0.50178973 & 21.995349 & 1.0296478 & 0.8428 \\
\hline
\end{tabular}
Given the information above what would be the appropriate decision? (Let Alpha $=0.05$ )
Fail to reject the alternative hypothesis $\left(\mathrm{H}_{\mathrm{A}}\right)$
Reject the alternative hyopthesis $\left(\mathrm{H}_{\mathrm{A}}\right)$
Fail to reject the null hypothesis $\left(\mathrm{H}_{0}\right)$
Reject the null hypothesis $\left(\mathrm{H}_{0}\right)$
Answer
\(\boxed{\text{Final Answer: Fail to reject the null hypothesis } \left(\mathrm{H}_{0}\right)}\)
Step 1 ：The null hypothesis is that the mean enjoyment rating for stories with spoilers is the same as the mean enjoyment rating for the original stories. The alternative hypothesis is that the mean enjoyment rating for stories with spoilers is less than the mean enjoyment rating for the original stories.
Step 2 ：The p-value is the probability of obtaining the observed data (or data more extreme) if the null hypothesis is true.
Step 3 ：If the p-value is less than the significance level (alpha), we reject the null hypothesis in favor of the alternative hypothesis. If the p-value is greater than alpha, we fail to reject the null hypothesis.
Step 4 ：In this case, the p-value is 0.8428 and alpha is 0.05. Since the p-value is greater than alpha, we fail to reject the null hypothesis.
Step 5 ：\(\boxed{\text{Final Answer: Fail to reject the null hypothesis } \left(\mathrm{H}_{0}\right)}\)
