Question

Math

Posted 3 months ago

```
A city planner will consider adding electric vehicle charging stations if she is convinced that more than $15 \%$ of residents would use them. She will survey a sample of residents to test the hypotheses $H_{0}: p \leq 15 \%$ vs. $H_{\mathrm{a}}: p>15 \%$, where $p$ is the true proportion of residents that would use the stations.
Suppose that $H_{0}$ is actually true.
Which situation below would have the lowest probability of a Type I error?
Choose 1 answer:
(A) $n=200$ and $\alpha=0.01$
(B) $n=200$ and $\alpha=0.05$
(C) $n=300$ and $\alpha=0.10$
```

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Answer from Sia

Posted 3 months ago

Solution by Steps

step 1

Identify the definition of a Type I error in hypothesis testing

step 2

Recognize that the probability of a Type I error is denoted by the significance level $\alpha$

step 3

Compare the given significance levels $\alpha$ for each option to determine which has the lowest value

step 4

Note that the sample size $n$ does not affect the probability of a Type I error when the null hypothesis is true

step 5

Conclude that the option with the lowest significance level $\alpha$ will have the lowest probability of a Type I error

[1] Answer

(A) $n=200$ and $\alpha=0.01$

Key Concept

Type I Error Probability

Explanation

The probability of a Type I error in hypothesis testing is determined by the significance level $\alpha$, not by the sample size $n$. The option with the lowest $\alpha$ has the lowest probability of committing a Type I error.

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