A local fast-food restaurant serves buffalo wings. The restaurant's managers notice that they normally sell the following proportions of flavors for their wings: [tex]$20\%$[/tex] Spicy Garlic, [tex]$10\%$[/tex] Classic Medium, [tex]$20\%$[/tex] Teriyaki, [tex]$20\%$[/tex] Hot BBQ, and [tex]$30\%$[/tex] Asian Zing. After running a campaign to promote their nontraditional specialty wings, they want to know if the campaign has made an impact. The results after 10 days are listed in the following table. Is there sufficient evidence at the 0.01 level of significance to say that the promotional campaign has made any difference in the proportions of flavors sold?

\begin{tabular}{|c|c|}
\hline \multicolumn{2}{|c|}{Buffalo Wing Sales} \\
\hline & Number Sold \\
\hline Spicy Garlic & 112 \\
\hline Classic Medium & 76 \\
\hline Teriyaki & 109 \\
\hline Hot BBQ & 124 \\
\hline Asian Zing & 145 \\
\hline
\end{tabular}

Step 1 of 4: State the null and alternative hypotheses in terms of the expected proportion for each flavor.

[tex]$H_0: p_{SG} = 0.20, \quad p_{CM} = 0.10, \quad p_{T} = 0.20, \quad p_{BBQ} = 0.20, \quad p_{AZ} = 0.30$[/tex]

[tex]$H_a$[/tex]: There is a difference from the stated proportions.

Question

23-06-2024
Mathematics

Answered

A local fast-food restaurant serves buffalo wings. The restaurant's managers notice that they normally sell the following proportions of flavors for their wings: [tex]$20\%$[/tex] Spicy Garlic, [tex]$10\%$[/tex] Classic Medium, [tex]$20\%$[/tex] Teriyaki, [tex]$20\%$[/tex] Hot BBQ, and [tex]$30\%$[/tex] Asian Zing. After running a campaign to promote their nontraditional specialty wings, they want to know if the campaign has made an impact. The results after 10 days are listed in the following table. Is there sufficient evidence at the 0.01 level of significance to say that the promotional campaign has made any difference in the proportions of flavors sold?

\begin{tabular}{|c|c|}
\hline \multicolumn{2}{|c|}{Buffalo Wing Sales} \\
\hline & Number Sold \\
\hline Spicy Garlic & 112 \\
\hline Classic Medium & 76 \\
\hline Teriyaki & 109 \\
\hline Hot BBQ & 124 \\
\hline Asian Zing & 145 \\
\hline
\end{tabular}

Step 1 of 4: State the null and alternative hypotheses in terms of the expected proportion for each flavor.

[tex]$H_0: p_{SG} = 0.20, \quad p_{CM} = 0.10, \quad p_{T} = 0.20, \quad p_{BBQ} = 0.20, \quad p_{AZ} = 0.30$[/tex]

[tex]$H_a$[/tex]: There is a difference from the stated proportions.

Answer :

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GinnyAnswer GinnyAnswer · Answer 1 · 2024-06-23T19:42:05-04:00

Step 1 of 4: State the null and alternative hypotheses in terms of the expected proportion for each flavor.

Given the proportions observed in the normal sales distribution before the promotional campaign and the observed sales after the campaign, we need to establish hypotheses to test whether the campaign has significantly altered the proportions sold of each flavor.

The null hypothesis ([tex]$H_0$[/tex]) represents the expected distribution if the campaign had no effect, based on the historical proportions. The alternative hypothesis ([tex]$H_a$[/tex]) states that at least one of the proportions is different from what was expected after the campaign.

Null Hypothesis ([tex]$H_0$[/tex]):
- [tex]$H_0: p_{SG} = 0.20$[/tex]
- [tex]$p_{CM} = 0.10$[/tex]
- [tex]$p_{T} = 0.20$[/tex]
- [tex]$p_{BBQ} = 0.20$[/tex]
- [tex]$p_{AZ} = 0.30$[/tex]

Alternative Hypothesis ([tex]$H_a$[/tex]):
- There is a difference from the stated proportions (i.e., at least one [tex]$p_i$[/tex] is different).

We are now ready to proceed to the subsequent steps where we will calculate the test statistic, the expected frequencies, and determine whether to reject the null hypothesis based on the 0.01 level of significance.