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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

REM (rapid eye movement) sleep is sleep during which most dreams occur. Each night a person has both REM and non-REM sleep. However, it is thought that children have more REM sleep than adults (Reference: Secrets of Sleep by Dr. A. Borbely). Assume that REM sleep time is normally distributed for both children and adults. A random sample of \(n_{1}=10\) children (9 years old) showed that they had an average REM sleep time of \(\bar{x}_{1}=2.8\) hours per night. From previous studies, it is known that \(\sigma_{1}=0.5\) hour. Another random sample of \(n_{2}=10\) adults showed that they had an average REM sleep time of \(\bar{x}_{2}=2.1\) hours per night. Previous studies show that \(\sigma_{2}=0.7\) hour. Do these data indicate that, on average, children tend to have more REM sleep than adults? Use a \(1 \%\) level of significance.

Weatherwise magazine is published in association with the American Meteorological Society. Volume 46 , Number 6 has a rating system to classify Nor'easter storms that frequently hit New England states and can cause much damage near the ocean coast. A severe storm has an average peak wave height of \(16.4\) feet for waves hitting the shore. Suppose that a Nor'easter is in progress at the severe storm class rating. (a) Let us say that we want to set up a statistical test to see if the wave action (i.e., height) is dying down or getting worse. What would be the null hypothesis regarding average wave height? (b) If you wanted to test the hypothesis that the storm is getting worse, what would you use for the alternate hypothesis? (c) If you wanted to test the hypothesis that the waves are dying down, what would you use for the alternate hypothesis? (d) Suppose you do not know whether the storm is getting worse or dying out. You just want to test the hypothesis that the average wave height is different (either higher or lower) from the severe storm class rating. What would you use for the alternate hypothesis? (e) For each of the tests in parts (b), (c), and (d), would the area corresponding to the \(P\) -value be on the left, on the right, or on both sides of the mean? Explain your answer in each case.

Consider a hypothesis test of difference of means for two independent populations \(x_{1}\) and \(x_{2} .\) What are two ways of expressing the null hypothesis?
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