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

step 1 We have a test of population mean mu, given by no, h0m u equals 80 versus h a mu is less than 80

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

To test $H_{0}: \mu=80$ versus $H_{1}: \mu<80,$ a simple random sample of size $n=22$ is obtained from a population that is known to be normally distributed. (a) If $\bar{x}=76.9$ and $s=8.5,$ 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 $t$ -distribution 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 simple random sample of size $n=22$ is obtained from a population that is known to be normally distributed.
(a) If $\bar{x}=76.9$ and $s=8.5,$ 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 $t$ -distribution that depicts the critical region.
(d) Will the researcher reject the null hypothesis? Why?

Key Concepts

Critical Value and Rejection Region

The critical value is a cutoff point on the probability distribution used to determine whether the test statistic falls into the rejection region. The rejection region is defined by the level of significance (alpha), and if the test statistic exceeds this critical value, the null hypothesis is rejected.

t-distribution

The t-distribution is a probability distribution that arises when estimating the mean of a normally distributed population in situations where the sample size is small and the population variance is unknown. It has heavier tails than the normal distribution, which provides a more accurate reflection of the uncertainty in the estimate of the standard error.

t-test for Mean

The t-test for the mean is used when the population standard deviation is unknown and the sample size is small. It involves comparing the sample mean to the hypothesized population mean, using the sample standard deviation to estimate the variability of the data.

Hypothesis Testing

Hypothesis testing is the framework used to make decisions about a population parameter, based on sample data. It involves stating a null hypothesis (H0), which represents a baseline claim about the parameter, and an alternative hypothesis (H1), which represents a claim to be supported if the data provide sufficient evidence against H0.

Test Statistic

The test statistic is a numerical value calculated from the sample data that measures the degree of agreement between the data and the null hypothesis. In the context of a t-test, it is computed as the difference between the sample mean and the hypothesized mean, divided by the sample standard error.

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