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Chapter 9: Problem 2

To test \(\mu\) for an \(x\) distribution that is mound-shaped using sample size \(n \geq 30\), how do you decide whether to use the normal or Student's distribution?

Short Answer

Expert verified
Use the normal distribution with \(n \geq 30\).

Step by step solution

01

Understand the Distributions

Before deciding which statistical distribution to use for hypothesis testing, it is crucial to understand what the normal distribution and Student's t-distribution are. The normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. On the other hand, the Student's t-distribution is similar to the normal distribution but accounts for variability in data when the sample size is small.
02

Consider the Sample Size Constraint

In the problem, the given sample size is at least 30 (\(n \geq 30\)). This is an important factor because, according to the Central Limit Theorem, when the sample size is large (usually \(n \geq 30\) is considered sufficiently large), the sampling distribution of the sample mean will approximate a normal distribution regardless of the shape of the population distribution.
03

Decide on the Distribution to Use

Since the sample size \(n\) is greater than or equal to 30, we can generally use the normal distribution (\(z\)-distribution). The reason is that with a large sample size, the sample mean will have a distribution that closely approximates the normal distribution. Student's t-distribution is particularly useful for smaller sample sizes because it accounts for the additional uncertainty in the estimate of the standard deviation.

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Key Concepts

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

Normal Distribution
In statistics, the normal distribution is essential for analyzing data that naturally clusters around a mean. Imagine a bell curve鈥攊t鈥檚 exactly what a normal distribution looks like. Symmetrical and bell-shaped, this type of distribution shows that most data points occur near the average, with fewer instances happening as you move away from the center.

Key characteristics include:
  • A mean and a median that are equal, located at the center of the distribution.
  • A standard deviation that measures the spread or variability of the distribution.
  • A symmetric shape, meaning half of the data lies to the left of the mean and half to the right.

The normal distribution is often referenced in hypothesis testing. It allows statisticians to determine the probability of a sample statistic occurring by chance. When sample sizes are large, the sample mean will have a distribution that approximates the normal curve, enabling the usage of the normal distribution for hypothesis tests.
Student's t-distribution
The Student's t-distribution acts like a safety net when dealing with smaller sample sizes鈥攖ypically those with fewer than 30 observations. Unlike the normal distribution, the t-distribution is more spread out and has heavier tails. This reflects that with smaller samples, there is greater variability and uncertainty.

Here鈥檚 what makes the t-distribution distinctive:
  • It鈥檚 symmetrical and bell-shaped, similar to the normal distribution, but wider.
  • The degrees of freedom (closely related to sample size) significantly impact its shape. Fewer degrees of freedom create a flatter and wider distribution, accounting for more variability.
  • As sample size increases, the t-distribution approaches the normal distribution.

This distribution is handy for hypothesis testing under uncertain conditions of small samples. It helps in estimating the population mean when the standard deviation is unknown and accommodates the additional uncertainty by providing wider confidence intervals and limiting the risk of incorrect conclusions.
Central Limit Theorem
The Central Limit Theorem (CLT) is a fundamental concept in statistics that plays a pivotal role in hypothesis testing. Simply put, the CLT states that regardless of the original population distribution鈥檚 shape, the distribution of the sample mean will tend to be normal, provided the sample size is sufficiently large (typically, this means an n of at least 30).

Key insights from the CLT include:
  • Once sample sizes reach a reasonable size, the sample means will closely follow a normal distribution.
  • This allows us to use the normal distribution to make inferences about the population mean, even if the population is not normally distributed.
  • The theorem validates using the normal distribution in hypothesis testing and confidence intervals when handling sufficiently large samples.

The CLT is the backbone of justifying the use of the normal distribution for large sample sizes in statistical analyses. It reassures us that with a sufficiently large sample, we can use metrics like the sample mean and apply statistical testing with confidence, knowing that the outcomes align with a normal distribution.

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Most popular questions from this chapter

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 had an average thickness of \(\mu=67 \mathrm{~cm}\) (Source: Avalancbe 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 \(\mathrm{cm}\) ): \(\begin{array}{lllllll}59 & 51 & 76 & 38 & 65 & 54 & 49\end{array}\) 62 \(\begin{array}{llllllll}68 & 55 & 64 & 67 & 63 & 74 & 65 & 79\end{array}\) i. Use a calculator with mean and standard deviation keys to verify that \(\bar{x} \approx 61.8 \mathrm{~cm}\) and \(s \approx 10.6 \mathrm{~cm} .\) ii. Assume the slab thickness has an approximately normal distribution. Use \(1 \%\) level of significance to test the claim that the mean slab thickness in the Vail region is different from that in Canada.

The following is based on information taken from Winter Wind Studies in Rocky Mountain National Park, by D. E. Glidden (Rocky Mountain Nature Association). At five weather stations on Trail Ridge Road in Rocky Mountain National Park, the peak wind gusts (in miles per hour) for January and April are recorded below. \(\begin{array}{l|ccccc} \hline \text { Weather Station } & 1 & 2 & 3 & 4 & 5 \\ \hline \text { January } & 139 & 122 & 126 & 64 & 78 \\ \hline \text { April } & 104 & 113 & 100 & 88 & 61 \\ \hline \end{array}\) Does this information indicate that the peak wind gusts are higher in January than in April? Use \(\alpha=0.01\).

Harper's Index reported that \(80 \%\) of all supermarket prices end in the digit 9 or \(5 .\) Suppose you check a random sample of 115 items in a supermarket and find that 88 have prices that end in 9 or \(5 .\) Does this indicate that less than \(80 \%\) of the prices in the store end in the digits 9 or 5 ? Use \(\alpha=0.05\).

In her book Red Ink Behaviors, Jean Hollands reports on the assessment of leading Silicon Valley companies regarding a manager's lost time due to inappropriate behavior of employees. Consider the following independent random variables. The first variable \(x_{1}\) measures manager's hours per week lost due to hot tempers, flaming e-mails, and general unproductive tensions: $$\begin{array}{llllllll} x_{1}: & 1 & 5 & 8 & 4 & 2 & 4 & 10 \end{array}$$ The variable \(x_{2}\) measures manager's hours per week lost due to disputes regarding technical workers' superior attitudes that their colleagues are "dumb and dispensable": $$\begin{array}{lllllllll} x_{2}: & 10 & 5 & 4 & 7 & 9 & 4 & 10 & 3 \end{array}$$ i. Use a calculator with sample mean and standard deviation keys to verify that \(\bar{x}_{1} \approx 4.86, s_{1} \approx 3.18, \bar{x}_{2}=6.5\), and \(s_{2} \approx 2.88\). ii. Does the information indicate that the population mean time lost due to hot tempers is different (either way) from population mean time lost due to disputes arising from technical workers' superior attitudes? Use \(\alpha=0.05\). Assume that the two lost-time population distributions are mound-shaped and symmetric.

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. What is 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) State your conclusion in the context of the application. Bill Alther is a zoologist who studies Anna's hummingbird (Calypte anna). (Reference: Hummingbirds, K. Long, W. Alther.) Suppose that in a remote part of the Grand Canyon, a random sample of six of these birds was caught, weighed, and released. The weights (in grams) were \(\begin{array}{llllll}3.7 & 2.9 & 3.8 & 4.2 & 4.8 & 3.1\end{array}\) The sample mean is \(\bar{x}=3.75\) grams. Let \(x\) be a random variable representing weights of Anna's hummingbirds in this part of the Grand Canyon. We assume that \(x\) has a normal distribution and \(\sigma=0.70\) gram. It is known that for the population of all Anna's hummingbirds, the mean weight is \(\mu=4.55\) grams. Do the data indicate that the mean weight of these birds in this part of the Grand Canyon is less than \(4.55\) grams? Use \(\alpha=0.01\).
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