Consider the following hypothesis test. H0: μ≥10 Ha: μ<10 The sample size is 120 and

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Consider the following hypothesis test. H0: μ≥10 ...

Consider the following hypothesis test. $$ \begin{aligned} & H_0: \mu \geq 10 \\ & H_{\mathrm{a}}: \mu<10 \end{aligned} $$ The sample size is 120 and the population standard deviation is 5 . Use $\alpha=.05$. If the actual population mean is 9 , the probability of a Type Il error is .2912. Suppose the researcher wants to reduce the probability of a Type II error to .10 when the actual population mean is 9 . What sample size is recommended?

Consider the following hypothesis test.
$$
\begin{aligned}
& H_0: \mu \geq 10 \\
& H_{\mathrm{a}}: \mu<10
\end{aligned}
$$
The sample size is 120 and the population standard deviation is 5 . Use $\alpha=.05$. If the actual population mean is 9 , the probability of a Type Il error is .2912. Suppose the researcher wants to reduce the probability of a Type II error to .10 when the actual population mean is 9 . What sample size is recommended?

Key Concepts

Type I Error

A Type I error occurs when the null hypothesis is wrongly rejected when it is actually true. The probability of making this error is denoted by the significance level, alpha, which is chosen by the researcher to control the risk of falsely declaring a significant effect.

Type II Error

A Type II error happens when the null hypothesis is not rejected even though it is false. This error is quantified by beta, the probability of failing to detect a true effect. Reducing beta increases the test’s sensitivity to detect an actual difference, often by adjusting factors such as sample size.

Sample Size Determination

Sample size determination involves calculating the number of observations required to achieve a desired level of statistical power while controlling for Type I and II errors. Increasing the sample size typically reduces the standard error of the mean, thereby making it easier to detect a true effect and reducing the probability of a Type II error.

Power Analysis

Power analysis is the process of determining the likelihood that a test will detect a true effect when it exists. The power of a test is defined as 1-beta, where beta is the probability of a Type II error. By specifying a desired power level, researchers can compute the necessary sample size to reliably detect differences of practical significance.

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample of data to infer that a certain condition is true for the entire population. It involves formulating a null hypothesis, which is a statement of no effect or status quo, and an alternative hypothesis, which represents the effect or difference one expects to detect.

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