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Chapter 9: Problem 4

Explain which of the following is a two-tailed test, a left-tailed test, or a right-tailed test. a. \(H_{0}: \mu=12, H_{1}: \mu<12\) b. \(H_{0}: \mu \leq 85, H_{1}: \mu>85\) c. \(H_{0}: \mu=33, H_{1}: \mu \neq 33\) Show the rejection and nonrejection regions for each of these cases by drawing a sampling distribution curve for the sample mean, assuming that it is normally distributed.

Short Answer

Expert verified
a. is a left-tailed test with rejection region <\( \mu=12 \) and non-rejection region >\( \mu=12 \). b. is a right-tailed test where rejection region is >\( \mu=85 \) and non-rejection region <\( \mu=81 \). c. is a two-tailed test with rejection regions \( \mu=33 \) and non-rejection region around \( \mu=33 \).

Step by step solution

01

Identify the kind of test from hypothesis a.

The testing hypothesis \(H_{1}: \mu<12\) suggests that values less than 12 are significant. Thus, it indicates a left-tailed test.
02

Identify rejection and non-rejection regions for hypothesis a.

For a left-tailed test, the rejection region is to the left of the critical value on the curve. So anything less than \(\mu=12\) is the rejection region while anything greater than or equal to \(\mu=12\) is the non-rejection region.
03

Identify the kind of test from hypothesis b.

The testing hypothesis \(H_{1}: \mu>85\) suggests that values greater than 85 is significant. Thus, it indicates a right-tailed test.
04

Identify rejection and non-rejection regions for hypothesis b.

For a right-tailed test, the rejection region is to the right of the critical value on the curve. So anything greater than \(\mu=85\) is the rejection region while anything less than or equal to \(\mu=85\) is the non-rejection region.
05

Identify the kind of test from hypothesis c.

The testing hypothesis \(H_{1}: \mu \neq 33\) suggests significant values are different from 33. Thus, it indicates a two-tailed test.
06

Identify rejection and non-rejection regions for hypothesis c.

For a two-tailed test, rejection regions are on both tails of the curve. Anything less than or greater than \(\mu=33\) are the rejection regions while the region around \(\mu=33\) is the non-rejection region.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Two-Tailed Test
A two-tailed test is a statistical method used to determine if there is a significant difference in either direction, both less than or greater than, from a specific hypothesized value. This type of test is used when the alternative hypothesis (denoted as \(H_1\)) is predicting that the parameter of interest is not equal to a certain value, for example, \(H_1: \mu eq 33\).

Here are some important aspects to remember about two-tailed tests:
  • The rejection regions are in both tails of the sampling distribution curve.
  • You reject the null hypothesis (\(H_0\)) if the test statistic falls into these outer regions.
  • This test is useful when you are open to deviations in either direction, not just one.
Left-Tailed Test
In hypothesis testing, a left-tailed test is used when our alternative hypothesis is specific to the mean being less than a certain number. For instance, with \(H_1: \mu < 12\), the test is focused on the possibility of the mean being significantly lower than 12.

Let's delve into some details:
  • The rejection region is to the left of the critical value on the distribution curve.
  • If your computed test statistic lands in this left region, you reject the null hypothesis.
  • This test is helpful when you are specifically interested in detecting a decrease or a drop in the parameter.
Right-Tailed Test
A right-tailed test applies when the alternative hypothesis indicates values greater than the null hypothesis. For example, with \(H_1: \mu > 85\), you aim to discover if the mean is meaningfully higher than 85.

Critical features of a right-tailed test include:
  • The rejection region is located to the right of the critical value.
  • You will reject the null hypothesis if your test statistic falls in this right-hand region.
  • This type of test is best when evaluating if a parameter exceeds a certain baseline.
Rejection Region
In any hypothesis test, deciding the rejection region is crucial. It's the part of the distribution curve where the values lead you to reject the null hypothesis.

Depending on the test type:
  • For a left-tailed test, the rejection region is to the extreme left.
  • For a right-tailed test, it moves to the extreme right.
  • In a two-tailed test, both extremes, left and right, house the rejection regions.
The boundaries of these regions are determined by the critical value.
Critical Value
The critical value is a point on the distribution curve that defines the threshold for the rejection region. It's crucial in identifying whether to reject \(H_0\) or not.
  • This value depends on the chosen level of significance, commonly denoted by \( \alpha \).
  • For instance, if \( \alpha = 0.05\), the critical values might be calculated for both tails, considering a two-tailed test scenario.
  • In left or right-tailed tests, the critical value would be accounted for on one side of the curve only.
Choosing the correct critical value helps in effectively making decisions based on statistical tests.

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Most popular questions from this chapter

Consider the following null and alternative hypotheses: $$ H_{0}: \mu=60 \text { versus } \quad H_{1}: \mu>60 $$ Suppose you perform this test at \(\alpha=.01\) and fail to reject the null hypothesis. Would you state that the difference between the hypothesized value of the population mean and the observed value of the sample mean is "statistically significant" or would you state that this difference is "statistically not significant?" Explain.

The administrative office of a hospital claims that the mean waiting time for patients to get treatment in its emergency ward is 25 minutes. A random sample of 16 patients who received treatment in the emergency ward of this hospital produced a mean waiting time of \(27.5\) minutes with a standard deviation of \(4.8\) minutes. Using a \(1 \%\) significance level, test whether the mean waiting time at the emergency ward is different from 25 minutes. Assume that the waiting times for all patients at this emergency ward have a normal distribution.

According to the National Association of Colleges and Employers, the average starting salary of 2014 college graduates with a bachelor's degree was \(\$ 45,473\) (www.naceweb.org). A random sample of 1000 recent college graduates from a large city showed that their average starting salary was \(\$ 44,930\). Suppose that the population standard deviation for the starting salaries of all recent college graduates from this city is \(\$ 7820\). a. Find the \(p\) -value for the test of hypothesis with the alternative hypothesis that the average starting salary of recent college graduates from this city is less than \(\$ 45,473 .\) Will you reject the null hypothesis at \(\alpha=.01 ?\) Explain. What if \(\alpha=.025 ?\) b. Test the hypothesis of part a using the critical-value approach. Will you reject the null hypothesis at \(\alpha=.01 ?\) What if \(\alpha=.025 ?\)

\(\quad\) Consider \(H_{0}: \mu=100\) versus \(H_{1}: \mu \neq 100\). a. A random sample of 64 observations produced a sample mean of \(98 .\) Using \(\alpha=.01\), would you reject the null hypothesis? The population standard deviation is known to be \(12 .\) b. Another random sample of 64 observations taken from the same population produced a sample mean of 104 . Using \(\alpha=.01\), would you reject the null hypothesis? The population standard deviation is known to be \(12 .\) Comment on the results of parts a and \(\mathrm{b}\).

Perform the following tests of hypothesis. \(\begin{array}{lll}\text { a. } H_{0}: \mu=285, & H_{1}: \mu<285, & n=55, \\ \bar{x}=267.80, & s=42.90, & a=.05 \\ \text { b. } H_{0} \cdot \mu=10.70, & H_{1}: \mu \neq 10.70, & n=47, \\\ \bar{x}=12.025, & s=4.90, & \alpha=.01 \\ \text { c. } H_{0} \cdot \mu=147,500, & H_{1}: \mu>147,500, & n=41, \\ \bar{x}=149,812, & s=22,972, & a=.10\end{array}\)
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