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Chapter 10: Problem 9

To test \(H_{0}: \mu=105\) versus \(H_{1}: \mu \neq 105,\) a simple random sample of size \(n=35\) is obtained. (a) Does the population have to be normally distributed to test this hypothesis by using the methods presented in this section? Why? (b) If \(\bar{x}=101.9\) and \(s=5.9,\) compute the test statistic. (c) Draw a \(t\) -distribution with the area that represents the \(P\) -value shaded. (d) Approximate and interpret the \(P\) -value. (e) If the researcher decides to test this hypothesis at the \(\alpha=0.01\) level of significance, will the researcher reject the null hypothesis? Why?

Short Answer

Expert verified
No, the population does not need to be normally distributed because the sample size is large. The test statistic is approximately -3.47. A t-distribution with 34 degrees of freedom will show the critical regions. The p-value is approximately 0.0015. At the 0.01 significance level, the null hypothesis is rejected.

Step by step solution

01

Determine if the population must be normally distributed (part a)

To use the methods of hypothesis testing presented in this section, the population from which the sample is drawn does not need to be normally distributed if the sample size is sufficiently large. According to the Central Limit Theorem, a sample size greater than 30 is generally considered large enough for the sampling distribution of the mean to be approximately normal. Since the sample size is 35, it is sufficiently large, so the population does not need to be normally distributed.
02

Calculate the test statistic (part b)

To calculate the test statistic, use the formula for the t-test for a single sample mean:

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Key Concepts

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

Population Distribution
Population distribution refers to the way in which values of a variable are spread out across a population. In hypothesis testing, the distribution of the population can affect the methods used for analysis.
However, if the sample size is large enough, typically greater than 30, we can rely on the Central Limit Theorem. This makes the sampling distribution approximate a normal distribution, regardless of the shape of the population distribution.
In our given exercise, since the sample size is 35, which is greater than 30, we do not need the population to be normally distributed. Thus, hypothesis testing methods such as the t-test are appropriate for this scenario.
Central Limit Theorem
The Central Limit Theorem (CLT) is a fundamental principle in statistics. It states that the sampling distribution of the sample mean will be approximately normally distributed if the sample size is large enough, typically greater than 30, regardless of the population's distribution.
In simpler terms, as our sample size increases, our sample averages will form a normal distribution, even if the original variable is not normally distributed.
In our exercise, since the sample size is 35, we can apply the CLT. This allows us to use methods like the t-test even without knowing the exact distribution of the original population.
t-test Calculation
The t-test is a statistical test used to compare the sample mean to the population mean when the population standard deviation is unknown. In our exercise, we calculate the test statistic using the formula: \[ t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}} \] where:
  • \(\bar{x}\) is the sample mean, which is 101.9
  • \(\mu\) is the population mean under the null hypothesis, which is 105
  • \(s\) is the sample standard deviation, which is 5.9
  • \(n\) is the sample size, which is 35
By plugging in these values, we get the test statistic \[ t = \frac{101.9 - 105}{\frac{5.9}{\sqrt{35}}} = -3.25 \] This value tells us how much our sample mean deviates from the population mean in units of the standard error.
P-value Interpretation
The P-value is a measure that helps us determine the significance of our test results. It represents the probability of observing a test statistic as extreme as, or more extreme than, the one obtained under the null hypothesis.
In our scenario, we need to look up the calculated t-value (-3.25) in a t-distribution table to find the P-value. Smaller P-values indicate stronger evidence against the null hypothesis.
For the given t-value, the P-value turns out to be approximately 0.002. This small P-value suggests that there is a very low probability of obtaining such a sample mean if the null hypothesis were true, leading us to believe the null hypothesis might not hold.
Significance Level
The significance level (\(\alpha\)) is a threshold set by the researcher to decide whether to reject the null hypothesis. It represents the maximum probability of making a Type I error, which is rejecting a true null hypothesis.
Common significance levels are 0.05, 0.01, and 0.10.
In the exercise, the significance level is set at \(\alpha = 0.01\).
Since our calculated P-value (0.002) is less than the significance level (0.01), we reject the null hypothesis. This means the sample provides strong evidence that the population mean is not 105.

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Most popular questions from this chapter

According to the National Highway and Traffic Safety Administration, the proportion of fatal traffic accidents in the United States in which the driver had a positive blood alcohol concentration (BAC) is 0.36. Suppose a random sample of 105 traffic fatalities in the state of Hawaii results in 51 that involved a positive BAC. Does the sample evidence suggest that Hawaii has a higher proportion of traffic fatalities involving a positive BAC than the United States at the \(\alpha=0.05\) level of significance?

Suppose we are testing the hypothesis \(H_{0}: p=0.65\) versus \(H_{1}: p \neq 0.65\) and we find the \(P\) -value to be \(0.02 .\) Explain what this means. Would you reject the null hypothesis? Why?

In August \(2002,47 \%\) of parents with children in grades \(\mathrm{K}-12\) were satisfied with the quality of education the students receive. A recent Gallup poll found that 437 of 1013 parents with children in grades \(\mathrm{K}-12\) were satisfied with the quality of education the students receive. Construct a \(95 \%\) confidence interval to assess whether this represents evidence that parents' attitudes toward the quality of education in the United States has changed since August 2002

Yale University graduate student J. Kiley Hamlin conducted an experiment in which 16 ten-month-old babies were asked to watch a climber character attempt to ascend a hill. On two occasions, the baby witnesses the character fail to make the climb. On the third attempt, the baby witnesses either a helper toy push the character up the hill or a hinderer toy prevent the character from making the ascent. The helper and hinderer toys were shown to each baby in a random fashion for a fixed amount of time. The baby was then placed in front of each toy and allowed to choose which toy he or she wished to play with. In 14 of the 16 cases, the baby chose the helper toy. Source: J. Kiley Hamlin et al., "Social Evaluation by Preverbal Infants." Nature, Nov. 2007. (a) Why is it important to randomly expose the baby to the helper or hinderer toy first? (b) What would be the appropriate null and alternative hypotheses if the researcher is attempting to show that babies prefer helpers over hinderers? (c) Use the binomial probability formula to determine the \(P\) -value for this test. (d) In testing 12 six-month-old babies, all 12 preferred the helper toy. The \(P\) -value was reported as \(0.0002 .\) Interpret this result.

Student loan debt has reached record levels in the United States. In a random sample of 100 individuals who have student loan debt, it was found the mean debt was 23,979 dollar with a standard deviation of 31,400 dollar . Data based on results from the Federal Reserve Bank of New York. (a) What do you believe is the shape of the distribution of student loan debt? Explain. (b) Use this information to estimate the mean student loan debt among all with such debt at the \(95 \%\) level of confidence. Interpret this result. (c) What could be done to increase the precision of the estimate?
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