An automobile mileage study tested the following bypocheses. Hypothesis H0: μ≥25 mp

step 1 So the first thing that we need to recall is how we determine our sample size, looking at Z at a

An automobile mileage study tested the following bypocheses....

An automobile mileage study tested the following bypocheses. $$ \begin{aligned} & \text { Hypothesis } \\ & H_0: \mu \geq 25 \mathrm{mpg} \\ & H_{\mathrm{a}}: \mu<25 \mathrm{mpg} \end{aligned} $$ Conclusion Manufacturer's claim supported Manufacturer's claim rejected; average mileage per gallon less than stated For $\sigma=3$ and a 02 level of significance, what sample size would be recommended if the researcher wants an $80 \%$ chance of detecting that $\mu$ is less than 25 miles per gallon when it is actually 24 ?

An automobile mileage study tested the following bypocheses.
$$
\begin{aligned}
& \text { Hypothesis } \\
& H_0: \mu \geq 25 \mathrm{mpg} \\
& H_{\mathrm{a}}: \mu<25 \mathrm{mpg}
\end{aligned}
$$
Conclusion
Manufacturer's claim supported
Manufacturer's claim rejected; average mileage per gallon less than stated
For $\sigma=3$ and a 02 level of significance, what sample size would be recommended if the researcher wants an $80 \%$ chance of detecting that $\mu$ is less than 25 miles per gallon when it is actually 24 ?

Key Concepts

Effect Size

Effect size measures the magnitude of the difference from the null hypothesis value that the test is designed to detect. It is a critical component in power analysis because a larger effect size generally requires a smaller sample size to be detected with the same power.

Z-test with Known Standard Deviation

When the population standard deviation is known, a z-test is used for hypothesis testing about the mean. This approach utilizes the normal distribution to compute critical values for significance and power calculations, which is crucial in determining the appropriate sample size for the study.

Sample Size Determination

Sample size determination involves calculating the number of observations needed to reliably detect a specified effect size under pre-set conditions such as significance level and power. Proper calculation ensures that the study is adequately powered to detect the effect, avoiding both false negatives and inefficient resource use.

Power of a Test

The power of a test, defined as 1 - ?, is the probability of correctly rejecting the null hypothesis when the alternative hypothesis is true. High power is desired so that true effects are detected, and it plays a key role in determining the required sample size.

One-Tailed Test

A one-tailed test is used when the research question specifies a particular direction of the effect. It tests whether the population parameter is either greater than or less than a given value, rather than simply being different. This directional focus increases the test's power to detect an effect in the specified direction.

Null and Alternative Hypotheses

The null hypothesis sets a baseline status regarding a population parameter, while the alternative hypothesis indicates the suspected effect or deviation. In hypothesis testing, correctly formulating these hypotheses is essential as they guide the decision process and the direction of the test.

Significance Level

The significance level, often denoted by ?, is the probability threshold for rejecting the null hypothesis when it is actually true (Type I error). Setting ? controls the strictness of the test and ensures that the chance of a false positive is kept within an acceptable limit.

Hypothesis Testing

Hypothesis testing is a fundamental method in statistics used to make decisions about population parameters. Researchers start with a null hypothesis (H0) that represents a default claim and an alternative hypothesis (Ha) that reflects their research interest, then use sample data to determine if there is enough evidence to reject H0 in favor of Ha.

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