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Chapter 4: Problem 36

4.36 True or false, Part I. Determine if the following statements are true or false, and explain your reasoning for statements you identify as false. (a) When comparing means of two samples where \(n_{1}=20\) and \(n_{2}=40,\) we can use the normal model for the difference in means since \(n_{2} \geq 30\). (b) As the degrees of freedom increases, the T distribution approaches normality. (c) We use a pooled standard error for calculating the standard error of the difference between means when sample sizes of groups are equal to each other.

Short Answer

Expert verified
(a) False; (b) True; (c) False.

Step by step solution

01

Understand the Problem Statement (a)

The statement claims that we can use the normal model for the difference in means if one of the sample sizes is greater than or equal to 30. The Central Limit Theorem suggests that a sample size of 30 or more is generally considered large enough for the sampling distribution of the mean to be approximately normal.
02

Evaluate Statement (a)

For the normal model to be appropriate, each sample doesn't necessarily need to be 30 or more unless the population distribution is severely non-normal. It's better to check each sample size individually. Since the sample size for the first group is only 20, the normal model might not be appropriate for this group unless the population distribution is normal.
03

Conclusion for Statement (a)

This statement is **false** because both sample sizes need to be large or the populations need to be normally distributed to use the normal model safely.
04

Understand the Problem Statement (b)

The statement suggests that as degrees of freedom increase, the T distribution becomes closer to a normal distribution.
05

Evaluate Statement (b)

The T distribution starts as broader and more spread out than the normal distribution, but as the degrees of freedom increase, it becomes narrower and more similar to the normal distribution.
06

Conclusion for Statement (b)

This statement is **true** because with more degrees of freedom, the T distribution indeed approaches normality.
07

Understand the Problem Statement (c)

The statement claims that a pooled standard error is used when sample sizes of groups are equal to each other.
08

Evaluate Statement (c)

A pooled standard error is used not necessarily when sample sizes are equal, but under the assumption of equal population variances regardless of sample sizes. Equal sample sizes help simplify variance calculations, but equal variance is the key assumption for pooling.
09

Conclusion for Statement (c)

This statement is **false** because pooled standard error relies on the assumption of equal variances from the populations, not just equal sample sizes.

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Key Concepts

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

Central Limit Theorem
The Central Limit Theorem (CLT) is a fundamental concept in statistics that explains how the distribution of sample means approximates a normal distribution as the sample size increases. This means that, regardless of the shape of the population distribution, the mean of a sufficiently large sample will be normally distributed.

According to the CLT:
  • The larger the sample size, the closer the sample mean distribution is to a normal distribution.
  • A sample size of 30 or more is often considered adequate to assume normality.
  • If the population distribution is severely non-normal, larger sample sizes may be necessary to achieve a normal distribution of the sample mean.
In exercise statement (a), the misunderstanding arises from interpreting the CLT application. While a sample size of 30 or more suggests approximating normality, both samples should reach this threshold if the population distribution is unknown or not normal, ensuring safe use of the normal model.
T distribution
The T distribution is a probability distribution used in statistical analysis, particularly in estimating population parameters when the sample size is small and/or the population standard deviation is unknown. It is similar to the normal distribution but has thicker tails, providing more room for variability from smaller sample sizes.

Key points about the T distribution:
  • It is used primarily when sample sizes are less than 30, or the population standard deviation is unknown.
  • It accounts for additional uncertainty introduced by small sample sizes.
  • The shape of the T distribution is determined by the degrees of freedom, which are related to the sample size.
Statement (b) is true because as the degrees of freedom increase, the T distribution becomes more like the normal distribution, implying increased confidence in estimates as sample sizes grow larger.
Pooled standard error
A pooled standard error is used when comparing means from two independent samples, particularly under the assumption that the population variances are equal. This approach combines the variances of the two samples, providing a more stable estimate of the variability between them.

Important considerations for pooled standard error:
  • It is not dependent on equal sample sizes, but rather on the assumption of equal variances across populations.
  • When variances are assumed equal, pooling improves the reliability of the standard error, even with unequal sample sizes.
  • A pooled standard error helps in gaining greater precision in calculating the standard error for the difference between means.
Statement (c) is false because the essential criterion for using a pooled standard error is the assumption of equal variances, not necessarily equal sample sizes.
Degrees of freedom
Degrees of freedom (df) is a statistical concept that describes the number of independent values or quantities that can vary in an analysis without breaking any given constraints. It is a crucial element, particularly in the context of the T distribution and hypothesis testing.

Understanding degrees of freedom involves:
  • In hypothesis testing, df are commonly calculated as the total sample size minus the number of estimated parameters.
  • In T tests, degrees of freedom impact the shape of the T distribution and help determine how closely it resembles a normal distribution.
  • As the degrees of freedom increase, particularly in larger samples, the T distribution closely approximates the normal distribution.
For our exercise, understanding that increased degrees of freedom result in the T distribution becoming increasingly normal helps clarify why statement (b) is considered true. More df indicates greater reliability in statistical inference.

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Most popular questions from this chapter

Does the Paleo diet work? The Paleo diet allows only for foods that humans typically consumed over the last 2.5 million years, excluding those agriculture-type foods that arose during the last 10,000 years or so. Researchers randomly divided 500 volunteers into two equal-sized groups. One group spent 6 months on the Paleo diet. The other group received a pamphlet about controlling portion sizes. Randomized treatment assignment was performed, and at the beginning of the study, the average difference in weights between the two groups was about 0. After the study, the Paleo group had lost on average 7 pounds with a standard deviation of 20 pounds while the control group had lost on average 5 pounds with a standard deviation of 12 pounds. (a) The \(95 \%\) confidence interval for the difference between the two population parameters (Paleo \- control) is given as (-0.891,4.891) . Interpret this interval in the context of the data. (b) Based on this confidence interval, do the data provide convincing evidence that the Paleo diet is more effective for weight loss than the pamphlet (control)? Explain your reasoning. (c) Without explicitly performing the hypothesis test, do you think that if the Paleo group had lost 8 instead of 7 pounds on average, and everything else was the same, the results would then indicate a significant difference between the treatment and control groups? Explain your reasoning.

Diamonds, Part I. Prices of diamonds are determined by what is known as the 4 Cs: cut, clarity, color, and carat weight. The prices of diamonds go up as the carat weight increases, but the increase is not smooth. For example, the difference between the size of a 0.99 carat diamond and a 1 carat diamond is undetectable to the naked human eye, but the price of a 1 carat diamond tends to be much higher than the price of a 0.99 diamond. In this question we use two random samples of diamonds, 0.99 carats and 1 carat, each sample of size \(23,\) and compare the average prices of the diamonds. In order to be able to compare equivalent units, we first divide the price for each diamond by 100 times its weight in carats. That is, for a 0.99 carat diamond, we divide the price by 99. For a 1 carat diamond, we divide the price by \(100 .\) The distributions and some sample statistics are shown below. \(^{37}\) Conduct a hypothesis test to evaluate if there is a difference between the average standardized prices of 0.99 and 1 carat diamonds. Make sure to state your hypotheses \begin{tabular}{|l|} \hline \\ \hline \end{tabular} clearly, check relevant conditions, and interpret your results in context of the data.

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Work hours and education, Part II. The General Social Survey described in Exercise 4.16 included random samples from two groups: US residents with a college degree and US residents without a college degree. For the 505 sampled US residents with a college degree, the average number of hours worked each week was 41.8 hours with a standard deviation of 15.1 hours. For those 667 without a degree, the mean was 39.4 hours with a standard deviation of 15.1 hours. Conduct a hypothesis test to check for a difference in the average number of hours worked for the two groups.

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