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Chapter 8: Problem 10

A random sample has 49 values. The sample mean is \(8.5\) and the sample standard deviation is \(1.5 .\) Use a level of significance of \(0.01\) to conduct a left-tailed test of the claim that the population mean is \(9.2\). (a) Is it appropriate to use a Student's \(t\) distribution? Explain. How many degrees of freedom do we use? (b) What are the hypotheses? (c) Compute the sample test statistic \(t\). (d) Estimate the \(P\) -value for the test. (e) Do we reject or fail to reject \(H_{0}\) ? (f) The results.

Short Answer

Expert verified
Reject the null hypothesis; the population mean is significantly less than 9.2.

Step by step solution

01

Determine Appropriateness of t-Distribution

Since the sample size is less than 30, using a Student's t-distribution is typically appropriate when the population standard deviation is unknown. However, our sample size is 49, which is greater than 30, and thus the sample can be considered large enough to use the t-distribution regardless of normality assumption. Degrees of freedom would be calculated as the sample size minus one, which is 48.
02

State the Hypotheses

The null hypothesis, \(H_0\), is that the population mean \( \mu = 9.2 \). The alternative hypothesis, \(H_a\), is that the population mean \( \mu < 9.2 \) since the test is left-tailed.
03

Compute the Test Statistic t

The formula for the t-statistic is \( t = \frac{\bar{x} - \mu}{s/\sqrt{n}} \), where \( \bar{x} = 8.5 \), \( \mu = 9.2 \), \( s =1.5 \), and \( n = 49 \). Plug these values into the formula:\[t = \frac{8.5 - 9.2}{1.5/\sqrt{49}} = \frac{-0.7}{1.5/7} = \frac{-0.7}{0.2143} \approx -3.27.\]
04

Estimate the P-value

Using a t-distribution table or calculator with 48 degrees of freedom, find the p-value for \( t = -3.27 \). The p-value is the probability that the t-statistic is less than -3.27. This p-value will be very small, less than 0.01.
05

Make a Conclusion about H0

Since the p-value is less than the level of significance \( \alpha = 0.01 \), we reject the null hypothesis \( H_0 \). This indicates there is sufficient evidence at the 0.01 significance level to support the claim that the population mean is less than 9.2.
06

State the Results

The results of the hypothesis test show there is sufficient evidence to conclude that the population mean is significantly less than 9.2 at the 0.01 level of significance.

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Key Concepts

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

Student's t-distribution
The Student's t-distribution is an essential concept in statistics, particularly when dealing with small sample sizes. It was developed by William Sealy Gosset under the pseudonym "Student."
Here鈥檚 why it's important:
  • When the population standard deviation is unknown, the t-distribution comes into play, especially for smaller samples (less than 30).
  • It is more spread out and has thicker tails than the normal distribution, which allows for greater variability in the data.
In our exercise, the sample size was 49. Although traditionally t-distributions are used for samples less than 30, they are still a reliable choice here due to unknown population standard deviation. This means that even with larger sample sizes, if we lack information about the population characteristics, the t-distribution allows for proper hypothesis testing.
Degrees of Freedom
Degrees of freedom is an important statistic that indicates the number of independent values in a dataset that can vary. When using the t-distribution, it is crucial to determine this number.
For our problem, the degrees of freedom (df) are calculated using the formula:
  • df = sample size - 1
In this case, the sample size is 49, so the degrees of freedom would be 48.
This is vital information because it affects the shape of the t-distribution. By knowing the degrees of freedom, we can precisely interpret how spread out the data might be and better understand the variability and reliability of the hypothesis test being performed.
P-value Estimation
Estimating the p-value is a fundamental part of hypothesis testing. The p-value helps us determine the strength and reliability of the evidence against the null hypothesis.
  • A low p-value (typically 鈮� 0.05) indicates strong evidence against the null hypothesis, so you may reject it.
  • A high p-value (> 0.05) suggests weak evidence against the null hypothesis, so you fail to reject it.
For our exercise, the computed t-statistic was approximately -3.27 using 48 degrees of freedom. This t-statistic is then used to find the corresponding p-value. Since we are conducting a left-tailed test, the p-value represents the probability of observing a t-statistic as extreme as -3.27 or more extreme. With a significance level of 0.01, and a very small p-value in this case, we find substantial evidence to reject the null hypothesis. Thus, interpreting the p-value appropriately guides our decision-making process and is key to completing the hypothesis test correctly.

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Most popular questions from this chapter

Consumer Reports stated that the mean time for a Chrysler Concorde to go from 0 to 60 miles per hour is \(8.7\) seconds. (a) If you want to set up a statistical test to challenge the claim of \(8.7\) seconds, what would you use for the null hypothesis? (b) The town of Leadville, Colorado, has an elevation over 10,000 feet. Suppose you wanted to test the claim that the average time to accelerate from 0 to 60 miles per hour is longer in Leadville (because of less oxygen). What would you use for the alternate hypothesis? (c) Suppose you made an engine modification and you think the average time to accelerate from 0 to 60 miles per hour is reduced. What would you use for the alternate hypothesis? (d) For each of the tests in parts (b) and (c), would the \(P\) -value area be on the left, on the right, or on both sides of the mean? Explain your answer in each case. For Problems \(19-24\), please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. Will you use a left-tailed, right-tailed, or two-tailed test? (b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. Compute the value of the sample test statistic. (c) Find (or estimate) the \(P\) -value. Sketch the sampling distribution and show the area corresponding to the \(P\) -value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level \(\alpha\) ? (e) Your conclusion in the context of the application.

Snow avalanches can be a real problem for travelers in the western United States and Canada. A very common type of avalanche is called the slab avalanche. These have been studied extensively by David McClung, a professor of civil engineering at the University of British Columbia. Slab avalanches studied in Canada have an average thickness of \(\mu=67\) (Source: Avalanche Handbook by D. McClung and P. Schaerer). The ski patrol at Vail, Colorado, is studying slab avalanches in its region. A random sample of avalanches in spring gave the following thicknesses (in \(\mathrm{cm})\) : \(\begin{array}{llllllll}59 & 51 & 76 & 38 & 65 & 54 & 49 & 62 \\ 68 & 55 & 64 & 67 & 63 & 74 & 65 & 79\end{array}\) i. Use a calculator with mean and standard deviation keys to verify that \(\bar{x}=61.8\) and \(s=10.6 \mathrm{~cm} .\) ii. Assume the slab thickness has an approximately normal distribution. Use a \(1 \%\) level of significance to test the claim that the mean slab thickness in the Vail region is different from that in Canada.

(a) For the same data and null hypothesis, is the \(P\) -value of a one-tailed test (right or left) larger or smaller than that of a two-tailed test? Explain. (b) For the same data, null hypothesis, and level of significance, is it possible that a one-tailed test results in the conclusion to reject \(H_{0}\) while a two-tailed test results in the conclusion to fail to reject \(H_{0}\) ? Explain. (c) For the same data, null hypothesis, and level of significance, if the conclusion is to reject \(H_{0}\) based on a two-tailed test, do you also reject \(H_{0}\) based on a one-tailed test? Explain. (d) If a report states that certain data were used to reject a given hypothesis, would it be a good idea to know what type of test (one-tailed or two-tailed) was used? Explain.

Consider a test for \(\mu\). If the \(P\) -value is such that you can reject \(H_{0}\) for \(\alpha=0.01\), can you always reject \(H_{0}\) for \(\alpha=0.05\) ? Explain.

Athabasca Fishing Lodge is located on Lake Athabasca in northern Canada. In one of its recent brochures, the lodge advertises that \(75 \%\) of its guests catch northern pike over 20 pounds. Suppose that last summer 64 out of a random sample of 83 guests did, in fact, catch northern pike weighing over 20 pounds. Does this indicate that the population proportion of guests who catch pike over 20 pounds is different from \(75 \%\) (either higher or lower)? Use \(\alpha=0.05\).
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