10.52 On testing Ho: p=14 H1: μ≠ 14 an α=0.05 level t -test is being considered. W

step 1 So let's solve this question. So in this question, we have been asked on testing H0 P14 H1, mu n

10.52 On testing Ho: p=14 H1: μ≠ 14 an α=0.05...

10.52 On testing $$\begin{array}{l}\text { Ho: } p=14 \\H_{1}: \quad \mu \neq 14\end{array}$$ an $\alpha=0.05$ level $t$ -test is being considered. What sample size is necessary in order that the probability is 0.1 of falsely failing to reject $H o$ when the true population mean differs from 14 by $0.5 ?$ From a preliminary sample we estimate $\sigma$ to be $1.25 .$

10.52 On testing
$$\begin{array}{l}\text { Ho: } p=14 \\H_{1}: \quad \mu \neq 14\end{array}$$
an $\alpha=0.05$ level $t$ -test is being considered. What sample size is necessary in order that the probability is 0.1 of falsely failing to reject $H o$ when the true population mean differs from 14 by $0.5 ?$ From a preliminary sample we estimate $\sigma$ to be $1.25 .$

Key Concepts

t-Test and Standard Error

The t-test is used to compare means when the population standard deviation is unknown and is instead estimated from the sample. The standard error, which is the estimated standard deviation of the sampling distribution of the mean, is critical in determining the t statistic and subsequently in making inferences about the population mean.

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 holds for the entire population. It involves formulating a null hypothesis, which represents a default position, and an alternative hypothesis, which is what the test seeks evidence for.

Type I and Type II Errors

In hypothesis testing, a Type I error occurs when the null hypothesis is wrongly rejected (a false positive), while a Type II error occurs when the test fails to reject a false null hypothesis (a false negative). Setting the significance level (alpha) controls the probability of a Type I error, whereas the probability of a Type II error (beta) is inversely related to the power of the test.

Test Power

Test power is defined as the probability of correctly rejecting a false null hypothesis (1 - beta). It reflects the sensitivity of the test to detect an effect when one truly exists. In sample size determination, ensuring adequate power is essential to minimize the risk of a Type II error.

Effect Size

Effect size quantifies the magnitude of the difference or change that the test is designed to detect, relative to the variability in the data. A larger effect size generally allows for detection with a smaller sample, whereas a smaller effect size requires a larger sample to achieve the same power.

Sample Size Determination

Sample size determination involves calculating the minimum number of observations required to achieve a desired power for the statistical test given the expected effect size, variability in the data, and the significance level. This ensures that the study is appropriately powered to detect meaningful differences or effects.

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