Question

Math

Posted 6 months ago

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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)
$\begin{array}{l}
H_{0}: p=0.3 \\
H_{\mathrm{a}}: p>0.3
\end{array}$
```

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Answer from Sia

Posted 6 months ago

Solution by Steps

step 1

Identify the correct null and alternative hypotheses for the significance test

step 2

The null hypothesis (H₀) should reflect the current belief or claim, which is that 30% of the customers are under 25 years old, so H₀: p = 0.3

step 3

The alternative hypothesis (Hₐ) should reflect the change the business wants to detect, which is an increase in the proportion of customers under 25 years old, so Hₐ: p > 0.3

step 4

Based on the hypotheses identified in steps 2 and 3, the appropriate set of hypotheses for the significance test is Option D

1 Answer

The correct set of hypotheses for the significance test is Option D: H₀: p = 0.3 and Hₐ: p > 0.3.

Key Concept

Hypothesis Testing for Proportions

Explanation

In hypothesis testing for proportions, the null hypothesis typically states that there is no change or difference from a known value, while the alternative hypothesis states the expected change or difference. In this case, the business wants to detect an increase, so the alternative hypothesis should reflect that the proportion is greater than the known value.

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