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. a. $\boldsymbol{H}_{\mathrm{a}}: \boldsymbol{\mu}-27 \mathrm{vs} . \boldsymbol{H}_{\mathrm{a}}: \mu<27 @ \mathrm{a}-\mathrm{0.05}, n=12, \mathrm{\sigma}=2.2$. b. $\boldsymbol{H}_{\mathrm{a}}: \boldsymbol{\mu}-5 \mathrm{~s}$ vs. $\boldsymbol{H}_{a}: \boldsymbol{\mu}+52 @ \mathrm{a}-0.05, n=6, \sigma$ unknown. C. $\boldsymbol{H}_{\mathrm{a}}: \boldsymbol{\mu}=-105 \mathrm{vs} . \boldsymbol{H}_{\mathrm{a}}: \boldsymbol{\mu}>-105 @ \mathrm{a}-\mathrm{0} .10, n=24, \sigma$ unknown. d. $\boldsymbol{H}_{\mathrm{D}}: \boldsymbol{\mu}-78.8$ vs. $\boldsymbol{H}_{\mathrm{a}}: \boldsymbol{\mu}+78.8 @ \mathrm{a}-\mathrm{0} .10, \mathrm{n}=8, \sigma=1.7$

Find the rejection region (for the standardized test statistic) for each hypothesis test based on the information given. The population is normally distributed.
a. $\boldsymbol{H}_{\mathrm{a}}: \boldsymbol{\mu}-27 \mathrm{vs} . \boldsymbol{H}_{\mathrm{a}}: \mu<27 @ \mathrm{a}-\mathrm{0.05}, n=12, \mathrm{\sigma}=2.2$.
b. $\boldsymbol{H}_{\mathrm{a}}: \boldsymbol{\mu}-5 \mathrm{~s}$ vs. $\boldsymbol{H}_{a}: \boldsymbol{\mu}+52 @ \mathrm{a}-0.05, n=6, \sigma$ unknown.
C. $\boldsymbol{H}_{\mathrm{a}}: \boldsymbol{\mu}=-105 \mathrm{vs} . \boldsymbol{H}_{\mathrm{a}}: \boldsymbol{\mu}>-105 @ \mathrm{a}-\mathrm{0} .10, n=24, \sigma$ unknown.
d. $\boldsymbol{H}_{\mathrm{D}}: \boldsymbol{\mu}-78.8$ vs. $\boldsymbol{H}_{\mathrm{a}}: \boldsymbol{\mu}+78.8 @ \mathrm{a}-\mathrm{0} .10, \mathrm{n}=8, \sigma=1.7$

Key Concepts

Critical Value

A critical value separates the rejection region from the acceptance region in the test statistic’s distribution. It is determined based on the chosen significance level and the type of test (one-tailed or two-tailed).

Known vs Unknown Population Standard Deviation

When the population standard deviation is known, the standardized test statistic is typically computed using the Z-distribution. When it is unknown and the sample size is small, the t-distribution is used to account for the additional uncertainty in estimating the standard deviation from the sample.

Standardized Test Statistic

A standardized test statistic transforms the sample statistic into a standard scale (usually normal or t distribution) by subtracting the hypothesized parameter value and dividing by the standard error. This allows comparisons to critical values from known distributions.

One-Tailed vs. Two-Tailed Tests

One-tailed tests specify a direction (either greater than or less than) for the alternative hypothesis, resulting in a rejection region in one tail of the distribution. Two-tailed tests consider deviations in both directions, creating rejection regions in both tails, and generally split the significance level between the two tails.

Significance Level (?)

The significance level is the threshold for determining when to reject the null hypothesis. It represents the probability of making a Type I error, where one wrongly rejects the null hypothesis when it is true. Commonly used values are 0.05 or 0.10.

Null and Alternative Hypotheses

The null hypothesis (H?) is a statement of no effect or no difference and serves as the default assumption to be tested. The alternative hypothesis (H?) represents the effect or difference that the researcher is interested in detecting, and it indicates the direction of the test (e.g., less than, greater than, or not equal to).

Hypothesis Testing

This is a statistical procedure used to decide whether there is enough evidence in a sample to infer that a certain condition holds for the entire population. It involves setting up a null hypothesis and an alternative hypothesis and then using data to decide if the observed effect is statistically significant.

Rejection Region

This is the set of all values of the test statistic that cause the null hypothesis to be rejected. The boundaries of this region are determined by the significance level and the distribution of the test statistic under the null hypothesis.

Transcript

00:01 In this exercise, we're going to be finding the rejection region for each of the hypothesis tests given below, and we're going to assume that the population is normally distributed.

00:18 In the first case, we are testing the null hypothesis that the population mean is equal to 27 versus the alternative hypothesis that the population mean is less than 27.

00:30 And the level of significance is 0 .05 and the sample size is 12 and the population standard deviation is 2 .2.

00:44 Now for this, since the population standard deviation is known, we're going to use the z statistic test statistic.

00:55 Now in this case, for us to determine the rejection region, we have to do.

01:03 Think about a left tail test here and since the left tail is a negative the critical value is going to be negative we check from the table and the value that corresponds to the 0 .05 levels of significance is that z is less than or equal to negative 1 .645.

01:37 For the second case, their alternative hypothesis is that the population mean is not equal to 52.

01:47 So that makes this a two -tailed test.

01:52 And in this case, we shall have to get two critical values, the positive and the negative critical value.

02:01 And since the population standard deviation is unknown, we shall use the t table so what we're looking for is plus or minus t for a two -tail test with a degrees of freedom as n minus one so in this case it's plus or minus t as 0 .0 25 and the degrees of freedom would be equal to five because that's six minus one so on the table you'll find that this corresponds to plus or minus 2 .571.

02:45 So that tells us that the value of t should be lessen or equal to negative 2 .571 or it should be greater than or equal to positive 2 .571.

03:00 Next, we are comparing, we are testing the hypothesis that the mean is equal to negative negative 105 versus the mean is greater than negative 105...

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