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Chapter 10: Problem 43

When are the samples considered large enough for the sampling distribution of the difference between two sample proportions to be (approximately) normal?

Short Answer

Expert verified
The samples are considered large enough for the sampling distribution of the difference between two proportions to be approximately normal when both np and n(1-p) are greater than or equal to 10 for each group's sample in your study.

Step by step solution

01

Understand Large Sample Sizes

The underlying principle governing the sample size is the Central Limit Theorem. According to this theory, for larger sample sizes, the distribution of sample means tends to a normal distribution. In other words, as the sample size increases, the shape of the distribution becomes more normal.
02

Compute the Minimum Sample Size

In terms of proportions, a rule of thumb is that the sample is considered 'large enough' if both np and n(1-p) are greater than or equal 10, where n is the sample size and p is the population proportion. This condition must be met by each group's sample in your study. Thus, if n1p1 > 10, n1(1-p1) > 10, n2p2 > 10, and n2(1-p2) > 10 (p1 and p2 are the known or estimated proportions of population 1 and 2, and n1 and n2 are the sizes of sample 1 and 2), the sample size is deemed 'large enough'.
03

Understand the Application

This rule ensures that the sampling distribution of the sample proportions will be approximately normal and can allow you to make inferences about the population based on your sample data. For example, if both conditions are met, one can use statistical techniques like z-tests to perform hypothesis tests and establish confidence intervals.

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Most popular questions from this chapter

A mail-order company has two warehouses, one on the West Coast and the second on the East Coast. The company's policy is to mail all orders placed with it within 72 hours. The company's quality control department checks quite often whether or not this policy is maintained at the two warehouses. A recently taken sample of 400 orders placed with the warehouse on the West Coast showed that 364 of them were mailed within 72 hours. Another sample of 300 orders placed with the warehouse on the East Coast showed that 279 of them were mailed within 72 hours. a. Construct a \(97 \%\) confidence interval for the difference between the proportions of all orders placed at the two warehouses that are mailed within 72 hours. b. Using a \(2.5 \%\) significance level, can you conclude that the proportion of all orders placed at the warehouse on the West Coast that are mailed within 72 hours is lower than the corresponding proportion for the warehouse on the East Coast?

Sixty-five percent of all male voters and \(40 \%\) of all female voters favor a particular candidate. A sample of 100 male voters and another sample of 100 female voters will be polled. What is the probability that at least 10 more male voters than female voters will favor this candidate?

The manager of a factory has devised a detailed plan for evacuating the building as quickly as possible in the event of a fire or other emergency. An industrial psychologist believes that workers actually leave the factory faster at closing time without following any system. The company holds fire drills periodically in which a bell sounds and workers leave the building according to the system. The evacuation time for each drill is recorded. For comparison, the psychologist also records the evacuation time when the bell sounds for closing time each day. A random sample of 36 fire drills showed a mean evacuation time of \(5.1\) minutes with a standard deviation of \(1.1\) minutes. A random sample of 37 days at closing time showed a mean evacuation time of \(4.2\) minutes with a standard deviation of \(1.0\) minute. a. Construct a \(99 \%\) confidence interval for the difference between the two population means. b. Test at a \(5 \%\) significance level whether the mean evacuation time is smaller at closing time than during fire drills. Assume that the evacuation times at closing time and during fire drills have equal but unknown population standard deviations.

According to a Bureau of Labor Statistics report released on March 25, 2015 , statisticians earn an average of \(\$ 84,010\) a year and accountants and auditors earn an average of \(\$ 73,670\) a year (www.bls. gov). Suppose that these estimates are based on random samples of 2000 statisticians and 1800 accountants and auditors. Further assume that the sample standard deviations of the annual earnings of these two groups are \(\$ 15,200\) and \(\$ 14,500\), respectively, and the population standard deviations are unknown and unequal for the two groups. a. Construct a \(98 \%\) confidence interval for the difference in the mean annual earnings of the two groups, statisticians and accountants and auditors. b. Using a \(1 \%\) significance level, can you conclude that the average annual earnings of statisticians is higher than that of accountants and auditors?

Quadro Corporation has two supermarket stores in a city. The company's quality control department wanted to check if the customers are equally satisfied with the service provided at these two stores. A sample of 380 customers selected from Supermarket 1 produced a mean satisfaction index of \(7.6\) (on a scale of 1 to 10,1 being the lowest and 10 being the highest) with a standard deviation of .75. Another sample of 370 customers selected from Supermarket II produced a mean satisfaction index of \(8.1\) with a standard deviation of \(.59\). Assume that the customer satisfaction indexes for the two supermarkets have unknown and unequal population standard deviations. a. Construct a \(98 \%\) confidence interval for the difference between the mean satisfaction indexes for all customers for the two supermarkets. b. Test at a \(1 \%\) significance level whether the mean satisfaction indexes for all customers for the two supermarkets are different. c. Suppose that the sample standard deviations were \(.88\) and \(.39\), respectively. Redo parts a and b. Discuss any changes in the results.
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