Find the rejection region (for the standardized test...

Find the rejection region (for the standardized test statistic) for each hypothesis test based on the information given. The population is normally distributed. Identify the test as left-tailed, right-tailed, or two-tailed. a. $\boldsymbol{H}_{\mathrm{a}}: \boldsymbol{\mu}--62$ vs. $\boldsymbol{H}_{a}: \mu \neq-62 @ a-0.005, n=8, \sigma$ unknown. b. $\boldsymbol{H}_{\mathrm{a}}: \boldsymbol{\mu}-73$ vs. $\boldsymbol{H}_{\mathrm{a}}: \boldsymbol{\mu}>73 @ \mathrm{a}-0.001, n=22, \sigma$ unknown. C. $\boldsymbol{H}_{\mathrm{a}}: \mu-1124$ vs. $\boldsymbol{H}_{\mathrm{a}}: \mu<1124 @ \mathrm{a}-0.001, n=21, \sigma$ unknown. d. $\boldsymbol{H}_{\mathrm{a}}: \mu-0.12$ vs. $\boldsymbol{H}_{a}: \mu+0.12 @ a-0.001, n=14, \sigma=0.026 .$

Find the rejection region (for the standardized test statistic) for each hypothesis test
based on the information given. The population is normally distributed. Identify the
test as left-tailed, right-tailed, or two-tailed.
a. $\boldsymbol{H}_{\mathrm{a}}: \boldsymbol{\mu}--62$ vs. $\boldsymbol{H}_{a}: \mu \neq-62 @ a-0.005, n=8, \sigma$ unknown.
b. $\boldsymbol{H}_{\mathrm{a}}: \boldsymbol{\mu}-73$ vs. $\boldsymbol{H}_{\mathrm{a}}: \boldsymbol{\mu}>73 @ \mathrm{a}-0.001, n=22, \sigma$ unknown.
C. $\boldsymbol{H}_{\mathrm{a}}: \mu-1124$ vs. $\boldsymbol{H}_{\mathrm{a}}: \mu<1124 @ \mathrm{a}-0.001, n=21, \sigma$ unknown.
d. $\boldsymbol{H}_{\mathrm{a}}: \mu-0.12$ vs. $\boldsymbol{H}_{a}: \mu+0.12 @ a-0.001, n=14, \sigma=0.026 .$

Key Concepts

Significance Level (Alpha)

The significance level (alpha) is the threshold probability used to decide whether to reject the null hypothesis. It represents the risk of making a Type I error—rejecting the null hypothesis when it is true. Alpha is used to determine the critical value(s) that define the rejection region.

Standardized Test Statistic

The standardized test statistic measures how far the sample statistic deviates from the null hypothesis parameter in terms of standard errors. It is used to compare the observed sample data to what is expected under the null hypothesis and is typically derived using the sample mean, hypothesized mean, and standard error.

t-Distribution and Degrees of Freedom

When the population standard deviation is unknown, especially with small sample sizes, the t-distribution is used instead of the normal distribution. The t-distribution accounts for additional uncertainty by incorporating degrees of freedom, which are typically calculated as the sample size minus one.

Rejection Region

The rejection region is the set of values for the test statistic that leads to rejecting the null hypothesis. Its determination depends on the significance level (alpha) and whether the test is one-tailed or two-tailed. When the calculated test statistic falls into this region, the null hypothesis is rejected.

One-Tailed vs Two-Tailed Tests

The decision between one-tailed (left-tailed or right-tailed) and two-tailed tests is based on the form of the alternative hypothesis. In a one-tailed test, the rejection region is entirely in one tail of the distribution, indicating a specific direction of deviation. In a two-tailed test, the rejection regions are split between both tails, reflecting deviations in either direction from the hypothesized parameter.

Please give Ace some feedback