We have provided a sample mean, sample standard deviation,...

We have provided a sample mean, sample standard deviation, and sample size. In each case, use the one-mean $t$-test to perform the required hypothesis test at the $5 \%$ significance level.
$$\bar{x}=23, s=4, n=24, H_{0}: \mu=22, H_{\mathrm{a}}: \mu \neq 22$$

Key Concepts

Significance Level

The significance level, often denoted by alpha (?), is the threshold for rejecting the null hypothesis. It represents the probability of committing a Type I error, which is the mistake of rejecting a true null hypothesis. In a two-tailed test, the significance level is split between the two tails of the t-distribution.

t-Statistic

The t-statistic is a ratio that measures the difference between the sample mean and the hypothesized population mean relative to the variability observed in the sample. It is calculated using the sample mean, the hypothesized mean, the sample standard deviation, and the sample size. The t-statistic follows a t-distribution under the null hypothesis, which is used to assess the evidence against the null.

Degrees of Freedom

Degrees of freedom in the context of a one-sample t-test generally refer to the number of independent values in the sample that are free to vary, calculated as the sample size minus one. This parameter is crucial in determining the shape of the t-distribution and affects the critical values used for hypothesis testing.

One-Sample t-Test

The one-sample t-test is a statistical method used to determine if the mean of a single sample differs significantly from a known or hypothesized population mean. This test is particularly useful when the population standard deviation is unknown and the sample size is relatively small, making the t-distribution an appropriate model for the test statistic.

Hypothesis Testing

Hypothesis testing is a formal procedure for testing whether a presumed claim about a population parameter is reasonable based on sample data. It involves setting up a null hypothesis (which usually represents no effect or no difference) and an alternative hypothesis, then using the sample data to decide whether to reject the null hypothesis at a predetermined significance level.

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