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Chapter 10: Problem 50

Perform the following tests of hypotheses, assuming that the populations of paired differences are normally distributed. a. \(H_{0}: \mu_{d}=0, \quad H_{1}: \mu_{d} \neq 0, \quad n=9, \quad \bar{d}=6.7, \quad s_{d}=2.5, \quad \alpha=.10\) b. \(H_{0}: \mu_{d}=0, \quad H_{1}: \mu_{d}>0, \quad n=22, \quad \bar{d}=14.8, \quad s_{d}=6.4, \quad \alpha=.05\) c. \(H_{0}: \mu_{d}=0, \quad H_{1}: \mu_{d}<0, \quad n=17, \quad \bar{d}=-9.3, \quad s_{d}=4.8, \quad \alpha=.01\)

Short Answer

Expert verified
In all three parts a, b, and c of the exercise, we reject the null hypothesis because our test statistic falls within the rejection region.

Step by step solution

01

Calculating the Test Statistics

The test statistic (t) for a paired T-test is calculated as follows: \( t = \frac{\bar{d} - \mu_{d}}{s_d/ \sqrt{n}} \)where - \( \bar{d} \) is the mean difference - \( \mu_{d} \) is the hypothesized population mean difference - \( s_d \) is the standard deviation of the differences - \( n \) is the number of differences Now, let's calculate the test statistic for each part:a. \( t = \frac{6.7 - 0}{2.5/ \sqrt{9}} = 8.04 \)b. \( t = \frac{14.8 - 0}{6.4/ \sqrt{22}} = 8.22 \)c. \( t = \frac{-9.3 - 0}{4.8/ \sqrt{17}} = -6.86
02

Determining the Rejection Regions

The rejection region (or critical region) is the range of values for which we reject the null hypothesis. The rejection region depends on the level of significance (\( \alpha \)) and the type of test (one-tailed or two-tailed).a. Since this is a two-tailed test (\( \mu_{d} \neq 0 \)), the rejection regions will be \( t < -t_{\alpha/2, n-1} \) and \( t > t_{\alpha/2, n-1} \).b. Since this is a one-tailed test (\( \mu_{d} > 0 \)), the rejection region will be \( t > t_{\alpha, n-1} \).c. Since this is a one-tailed test (\( \mu_{d} < 0 \)), the rejection region will be \( t < -t_{\alpha, n-1} \).
03

Comparing Test Statistic to the Rejection Region

Now we need to compare our computed test statistic to the critical value to determine whether we should reject the null hypothesis.a. With \( \alpha = 0.10 \), \( df = 9-1 \) and two-tailed test, the critical value is approximately \( \pm 1.86 \). Since 8.04 falls within the rejection region, we reject the null hypothesis.b. With \( \alpha = 0.05 \), \( df = 22-1 \) and right-tailed test, the critical value is approximately 1.72. Since 8.22 is greater than 1.72, we reject the null hypothesis.c. With \( \alpha = 0.01 \), \( df = 17-1 \) and left-tailed test, the critical value is approximately -2.62. Since -6.86 is less than -2.62, we reject the null hypothesis.

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Key Concepts

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

Hypothesis Testing
Hypothesis testing is a crucial process in statistics used to determine whether a specific claim about a population parameter is true. It essentially helps us make decisions based on sample data.
  • First, you start with a statement, the hypothesis, about a population parameter.
  • Then you collect data and analyze it to see if it supports this hypothesis.
  • Finally, you use statistical methods to determine the likelihood of observing such data if the hypothesis were true.
In paired t-tests, like our example, we are tasked with deciding whether the means of two dependent populations differ, often gauged through differences in paired observations.
Null Hypothesis
The null hypothesis, denoted as \(H_0\), is a statement that there is no effect or no difference. It serves as the default or status quo against which evidence is tested.
In our exercise, we assume the null hypothesis \(H_0: \mu_d = 0\), meaning that there is no mean difference between the paired samples.
Rejecting the null hypothesis suggests that there is significant evidence to support the alternative hypothesis, indicating a potential effect or difference.
Critical Value
The critical value is a point (or points) on the scale of the test statistic beyond which we either reject or fail to reject the null hypothesis. It essentially serves as a threshold in hypothesis testing.
To determine the critical value, we need to consider two main factors: the significance level (\(\alpha\)) and the degrees of freedom, which depend on the sample size.
  • In "a" where \(\alpha = 0.10\), our critical value is approximately \(\pm 1.86\).
  • In "b" with \(\alpha = 0.05\), it's around 1.72 for a right-tailed test.
  • For "c" with \(\alpha = 0.01\), the critical value is approximately -2.62 for a left-tailed test.
Each critical value marks the boundary of our decision criteria.
Significance Level
The significance level, represented by \(\alpha\), is the probability of rejecting the null hypothesis when it is true. It indicates the risk of a Type I error (false positive).
Setting a lower significance level means you'll require stronger evidence to reject the null hypothesis. Common values for \(\alpha\) include 0.01, 0.05, and 0.10.
In our example, different significance levels are used:
  • \(\alpha = 0.10\) in part "a", which means there's a 10% risk of a false positive.
  • \(\alpha = 0.05\) in part "b", indicating a 5% risk.
  • \(\alpha = 0.01\) in part "c", implying a stricter 1% chance of a Type I error.
Choosing \(\alpha\) appropriately is vital based on the context and consequences of the results.

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Most popular questions from this chapter

A June 2009 Gallup Poll asked a sample of Americans whether they trust specific groups or individuals when it comes to making recommendations about healthcare reform. Sixty percent of Democrats and \(68 \%\) of Republicans stated that they trust doctors' opinions about healthcare reform (Source: http://www.gallup.com/poll/120890/Healthcare-Americans-Trust-Physicians- Politicians.aspx). Suppose this survey included 340 Democrats and 306 Republicans. a. Make a \(90 \%\) confidence interval for the difference in the population proportions for the two groups of people. b. At the \(5 \%\) significance level, can you conclude that the proportion of all Democrats who trust doctors' opinions about healthcare reform differs from the proportion of all Republicans who trust doctors' opinions about healthcare reform?

Quadro Corporation has two supermarket stores in a city. The company's quality control department wanted to check if the customers are equally satisfied with the service provided at these two stores. A sample of 380 customers selected from Supermarket I produced a mean satisfaction index of \(7.6\) (on a scale of 1 to 10,1 being the lowest and 10 being the highest) with a standard deviation of 75 . Another sample of 370 customers selected from Supermarket II produced a mean satisfaction index of \(8.1\) with a standard deviation of \(.59 .\) Assume that the customer satisfaction index for each supermarket has unknown but same population standard deviation. a. Construct a \(98 \%\) confidence interval for the difference between the mean satisfaction indexes for all customers for the two supermarkets. b. Test at the \(1 \%\) significance level whether the mean satisfaction indexes for all customers for the two supermarkets are different.

Maine Mountain Dairy claims that its 8-ounce low-fat yogurt cups contain, on average, fewer calories than the 8-ounce low-fat yogurt cups produced by a competitor. A consumer agency wanted to check this claim. A sample of 27 such yogurt cups produced by this company showed that they contained an average of 141 calories per cup. A sample of 25 such yogurt cups produced by its competitor showed that they contained an average of 144 calories per cup. Assume that the two populations are normally distributed with population standard deviations of \(5.5\) and \(6.4\) calories, repectively. a. Make a \(98 \%\) confidence interval for the difference between the mean number of calories in the 8-ounce low-fat yogurt cups produced by the two companies. b. Test at the \(1 \%\) significance level whether Maine Mountain Dairy's claim is true. c. Calculate the \(p\) -value for the test of part b. Based on this \(p\) -value, would you reject the null hypothesis if \(\alpha=.005 ?\) What if \(\alpha=.025\) ?

A local college cafeteria has a self-service soft ice cream machine. The cafeteria provides bowls that can hold up to 16 ounces of ice cream. The food service manager is interested in comparing the average amount of ice cream dispensed by male students to the average amount dispensed by female students. A measurement device was placed on the ice cream machine to determine the amounts dispensed. Random samples of 85 male and 78 female students who got ice cream were selected. The sample averages were \(7.23\) and \(6.49\) ounces for the male and female students, respectively. Assume that the population standard deviations are \(1.22\) and \(1.17\) ounces, respectively. a. Let \(\mu_{1}\) and \(\mu_{2}\) be the population means of ice cream amounts dispensed by all male and female students at this college, respectively. What is the point estimate of \(\mu_{1}-\mu_{2} ?\) b. Construct a \(95 \%\) confidence interval for \(\mu_{1}-\mu_{2}\). c. Using the \(1 \%\) significance level, can you conclude that the average amount of ice cream dispensed by male college students is larger than the average amount dispensed by female college students? Use both approaches to make this test.

A July 2009 Pew Research Center survey asked a variety of science questions of independent random samples of scientists and the public at-large (http://people-press.org/report/528/). One of the questions asked was whether all parents should be required to vaccinate their children. The percentage of people answering "yes" to this question was \(69 \%\) of the general public and \(82 \%\) of scientists. Suppose that the survey included 110 members of the general public and 105 scientists. a. Construct a \(98 \%\) confidence interval for the difference between the two population proportions. b. Using the \(1 \%\) significance level, can you conclude that the percentage of the general public who feels that all parents should be required to vaccinate their children is less than the percentage of all scientists who feels that all parents should be required to vaccinate their children? Use the critical-value and \(p\) -value approaches. c. The actual sample sizes used in the survey were 2001 members of the general public and 1005 scientists. Repeat parts a and b using the actual sample sizes. Does your conclusion change in part b?
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