To test H0: μ=80 versus H1: μ<80, a random sample of size n=22 is obtained from a population tha

step 1 Alright, so first we want to calculate our t -statistic. Using our formula, we are going to take

To test H0: μ=80 versus H1: μ<80, a random sample of size...

To test $H_{0}: \mu=80$ versus $H_{1}: \mu<80,$ a random sample of size $n=22$ is obtained from a population that is known to be normally distributed with $\sigma=11$ (a) If the sample mean is determined to be $\bar{x}=76.9$ compute the test statistic. (b) If the researcher decides to test this hypothesis at the $\alpha=0.02$ level of significance, determine the critical value. (c) Draw a normal curve that depicts the critical region. (d) Will the researcher reject the null hypothesis? Why?

To test $H_{0}: \mu=80$ versus $H_{1}: \mu<80,$ a random sample of size $n=22$ is obtained from a population that is known to be normally distributed with $\sigma=11$
(a) If the sample mean is determined to be $\bar{x}=76.9$ compute the test statistic.
(b) If the researcher decides to test this hypothesis at the $\alpha=0.02$ level of significance, determine the critical value.
(c) Draw a normal curve that depicts the critical region.
(d) Will the researcher reject the null hypothesis? Why?

Key Concepts

Critical Region and Decision Rule

The critical region is the set of outcomes for which the null hypothesis is rejected. The decision rule in hypothesis testing involves comparing the computed test statistic to the critical value: if the test statistic falls into the critical region, the null hypothesis is rejected; otherwise, it is not rejected. This process formally guides the decision-making in hypothesis testing and quantifies the risk of error.

Z Test Statistic

A Z test statistic is used when the population variance is known and the distribution of the sample mean is normal. It measures how far the sample mean is from the population mean in units of the standard error. The test statistic is calculated using the formula z = (x? - ?0) / (?/?n), making it an essential tool in determining whether to reject the null hypothesis.

Significance Level and Critical Value

The significance level, denoted by alpha (?), represents the probability of making a Type I error, which is rejecting the null hypothesis when it is actually true. The critical value is a threshold derived from this significance level that defines the boundary of the rejection region. For a one-tailed test, this value is determined from the standard normal distribution corresponding to the specified ? level.

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample to infer that a certain condition is true for the entire population. It involves formulating a null hypothesis (H0), which represents no effect or a baseline condition, and an alternative hypothesis (H1) which represents the effect or condition to be tested against the null hypothesis.

Sampling Distribution of the Sample Mean

The sampling distribution of the sample mean refers to the probability distribution of all possible sample means from a population. When sampling from a normally distributed population, or when the sample size is large enough due to the Central Limit Theorem, the sample mean will also be normally distributed. This concept underpins many statistical inference procedures where known population standard deviation allows for the use of standard normal calculations.

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