A business knows that $30 \%$ of its customers are less than 25 years old. The business wants to increase this percentage, so they create a marketing campaign that targets this age group. They plan on taking a sample of customers after the marketing campaign is complete to test if the proportion of customers in that age group has significantly increased. Let $p$ represent the proportion of this business's customers who are less than 25 years old after the marketing campaign. Which of the following is an appropriate set of hypotheses for their significance test? Choose 1 answer: (A) $$ \begin{array}{l} H_{0}: p \neq 0.3 \ H_{\mathrm{a}}: p=0.3 \end{array} $$ (B) $$ \begin{array}{l} H_{0}: p=0.3 \ H_{\mathrm{a}}: p \neq 0.3 \end{array} $$ (C) $$ \begin{array}{l} H_{0}: p=0.3 \ H_{\mathrm{a}}: p0.3 \end{array} $$

Question

A business knows that $30 \%$ of its customers are less than 25 years old. The business wants to increase this percentage, so they create a marketing campaign that targets this age group. They plan on taking a sample of customers after the marketing campaign is complete to test if the proportion of customers in that age group has significantly increased.

Let $p$ represent the proportion of this business's customers who are less than 25 years old after the marketing campaign.

Which of the following is an appropriate set of hypotheses for their significance test?

Choose 1 answer:
(A)
$$
\begin{array}{l}
H_{0}: p \neq 0.3 \
H_{\mathrm{a}}: p=0.3
\end{array}
$$
(B)
$$
\begin{array}{l}
H_{0}: p=0.3 \
H_{\mathrm{a}}: p \neq 0.3
\end{array}
$$
(C)
$$
\begin{array}{l}
H_{0}: p=0.3 \
H_{\mathrm{a}}: p<0.3
\end{array}
$$
(D)
$$
\begin{array}{l}
H_{0}: p=0.3 \
H_{\mathrm{a}}: p>0.3
\end{array}
$$

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Identify the correct null hypothesis (H₀) for the situation. Since the business knows that 30% of its customers are less than 25 years old, the null hypothesis should state that the proportion after the campaign is the same, which is $p = 0.3$. ‖ ‖ step 2 ⋮ Determine the correct alternative hypothesis (Hₐ) for the situation. The business wants to increase this percentage, so the alternative hypothesis should reflect that the proportion is greater than the known 30%, which is $p > 0.3$. ‖ ‖ step 3 ⋮ Match the identified H₀ and Hₐ to the correct set of hypotheses among the given options. ‖ ***D*** ∻Key Concept∻ ⚹Hypothesis Testing for Proportions⚹ ∻Explanation∻ ⚹In hypothesis testing for proportions, the null hypothesis typically states that there is no change or difference (the status quo), while the alternative hypothesis states the expected change or difference based on the research question or claim. In this case, the business wants to test if the proportion of customers under 25 has increased, so the null hypothesis is $p = 0.3$ and the alternative hypothesis is $p > 0.3$.⚹◊What is the significance level typically set at for hypothesis tests in statistics?⍭ Generate me a similar question◊

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