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Chapter 10: Problem 2

Describe the sampling distribution of \(\bar{x}_{1}-\bar{x}_{2}\) for two independent samples when \(\sigma_{1}\) and \(\sigma_{2}\) are known and either both sample sizes are large or both populations are normally distributed. What are the mean and standard deviation of this sampling distribution?

Short Answer

Expert verified
The mean of the sampling distribution of the difference between means (\(\bar{x}_{1}-\bar{x}_{2}\)) is given by \(\mu_{\bar{x}_{1}-\bar{x}_{2}} = \mu_{1} - \mu_{2}\). The standard deviation is given by \(\sigma_{\bar{x}_{1}-\bar{x}_{2}} = \sqrt{\left(\frac{\sigma_{1}^{2}}{n_{1}}\right) + \left(\frac{\sigma_{2}^{2}}{n_{2}}\right)}\).

Step by step solution

01

State the properties of the sampling distribution

When the samples come from normally distributed populations, or when the sample sizes are large, the central limit theorem states that the sampling distribution of the difference of sample means (\(\bar{x}_{1}-\bar{x}_{2}\) is approximately normally distributed, regardless of the shape of the populations. The mean and standard deviation are used to describe the location and dispersion of this distribution respectively. The mean of this sampling distribution is \(\mu_{\bar{x}_{1}-\bar{x}_{2}} = \mu_{1} - \mu_{2}\), where \(\mu_{1}\) and \(\mu_{2}\) are the population means.
02

Calculate the mean of the sampling distribution

The mean of the sampling distribution of the difference between means is equal to the difference in population means. If we assume that \(\mu_{1}\) and \(\mu_{2}\) are the population means, then for the mean, \(\mu_{\bar{x}_{1}-\bar{x}_{2}}\), of the sampling distribution of the difference between means, we have \(\mu_{\bar{x}_{1}-\bar{x}_{2}} = \mu_{1} - \mu_{2}\)
03

Calculate the standard deviation of the sampling distribution

The standard deviation of difference between means is calculated by combining the standard deviations of the two samples. This is given by the square root of the sum of the variances divided by the respective sample sizes. If \(\sigma_{1}\) and \(\sigma_{2}\) are the known population standard deviations and \(n_{1}\) and \(n_{2}\) are the sample sizes, then the standard deviation, \(\sigma_{\bar{x}_{1}-\bar{x}_{2}}\), of the difference between sample means is calculated as \(\sigma_{\bar{x}_{1}-\bar{x}_{2}} = \sqrt{\left(\frac{\sigma_{1}^{2}}{n_{1}}\right) + \left(\frac{\sigma_{2}^{2}}{n_{2}}\right)}\).

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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 is a fundamental concept in statistics that enables us to make inferences about population parameters based on sample data. It states that the distribution of the sample mean will be approximately normally distributed, regardless of the shape of the population, provided the sample size is sufficiently large. This is particularly useful when working with differences of sample means, like \( \bar{x}_{1}-\bar{x}_{2} \).
  • Key takeaway: even if individual populations aren't normally distributed, the sampling distribution of the mean approaches normality with larger samples.
  • Practical implication: researchers can apply statistical methods that assume normality even if the original data isn't normally distributed.
The theorem emphasizes that the larger the sample size, the closer the sampling distribution of the mean will get to a normal distribution. This is why in practical applications, large samples are often preferred.
Difference of Sample Means
When comparing two independent samples, one significant focus is the difference of sample means, denoted as \( \bar{x}_{1}-\bar{x}_{2} \).This value reflects how much the average value of one sample deviates from the other. Understanding this concept helps answer questions about whether there is a significant difference between the two groups being studied.
  • This difference is particularly meaningful when analyzing experiments or surveys where comparison between two groups is necessary.
  • It provides insight into the effect size or magnitude of differences between populations.
By analyzing the difference of sample means, researchers can make more informed decisions and develop further statistical models to support or reject hypotheses.
Population Mean Difference
The population mean difference is akin to the difference of sample means, yet on a broader scale. It involves comparing the average values (\( \mu_{1} \) and \( \mu_{2} \)) of two entire populations.
  • This comparison allows us to estimate whether the two populations as a whole differ from one another in some meaningful way.
  • In any analysis involving sampling, understanding the true population difference helps to ensure accuracy and relevance of conclusions.
In practice, when the means of populations are assumed known or estimated well, we calculate the difference to gauge the effect size among the populations. This step is vital in fields like medical research or quality control, where understanding population differences can influence policy or future research direction.
Standard Deviation of Sample Means
In assessing the difference between sample means, another crucial concept is understanding the standard deviation of these means, known as \( \sigma_{\bar{x}_{1}-\bar{x}_{2}} \).It is calculated using the known standard deviations of the populations (\( \sigma_{1} \) and \( \sigma_{2} \)), and the sample sizes (\( n_{1} \) and \( n_{2} \)).
  • The formula is: \[ \sigma_{\bar{x}_{1}-\bar{x}_{2}} = \sqrt{\left(\frac{\sigma_{1}^{2}}{n_{1}}\right) + \left(\frac{\sigma_{2}^{2}}{n_{2}}\right)} \]
  • This measurement indicates the variability or spread of the sample mean differences from their expected value (population mean difference).
Having a clear understanding of this measurement helps in evaluating the reliability and precision of the sample mean differences. It serves as an indicator of how much the sample means can vary if the study were repeated multiple times, making it essential for hypothesis testing and confidence interval construction.

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Most popular questions from this chapter

The management of a supermarket chain wanted to investigate if the percentages of men and women who prefer to buy national brand products over the store brand products are different. A sample of 600 men shoppers at the company's supermarkets showed that 246 of them prefer to buy national brand products over the store brand products. Another sample of 700 women shoppers at the company's supermarkets showed that 266 of them prefer to buy national brand products over the store brand products. a. What is the point estimate of the difference between the two population proportions? b. Construct a \(98 \%\) confidence interval for the difference between the proportions of all men and all women shoppers at these supermarkets who prefer to buy national brand products over the store brand products. c. Testing at a \(1 \%\) significance level, can you conclude that the proportions of all men and all women shoppers at these supermarkets who prefer to buy national brand products over the store brand products are different?

The management at New Century Bank claims that the mean waiting time for all customers at its branches is less than that at the Public Bank, which is its main competitor. A business consulting firm took a sample of 200 customers from the New Century Bank and found that they waited an average of \(4.5\) minutes before being served. Another sample of 300 customers taken from the Public Bank showed that these customers waited an average of \(4.75\) minutes before being served. Assume that the standard deviations for the two populations are \(1.2\) and \(1.5\) minutes, respectively. a. Make a \(97 \%\) confidence interval for the difference between the two population means. b. Test at a \(2.5 \%\) significance level whether the claim of the management of the New Century Bank is true. c. Calculate the \(p\) -value for the test of part b. Based on this \(p\) -value, would you reject the null hypothesis if \(\alpha=.01\) ? What if \(\alpha=.05\) ?

Assuming that the two populations are normally distributed with unequal and unknown population standard deviations, construct a \(95 \%\) confidence interval for \(\mu_{1}-\mu_{2}\) for the following. $$ \begin{array}{lll} n_{1}=14 & \bar{x}_{1}=109.43 & s_{1}=2.26 \\ n_{2}=15 & \bar{x}_{2}=113.88 & s_{2}=5.84 \end{array} $$

A local college cafeteria has a self-service soft ice cream machine. The cafeteria provides bowls that can hold up to 16 ounces of ice cream. The food service manager is interested in comparing the average amount of ice cream dispensed by male students to the average amount dispensed by female students. A measurement device was placed on the ice cream machine to determine the amounts dispensed. Random samples of 85 male and 78 female students who got ice cream were selected. The sample averages were \(7.23\) and \(6.49\) ounces for the male and female students, respectively. Assume that the population standard deviations are \(1.22\) and \(1.17\) ounces, respectively. a. Let \(\mu_{1}\) and \(\mu_{2}\) be the population means of ice cream amounts dispensed by all male and all female students at this college, respectively. What is the point estimate of \(\mu_{1}-\mu_{2} ?\) b. Construct a \(95 \%\) confidence interval for \(\mu_{1}-\mu_{2}\). c. Using a \(1 \%\) significance level, can you conclude that the average amount of ice cream dispensed by all male college students is larger than the average amount dispensed by all female college students? Use both approaches to make this test.

Gamma Corporation is considering the installation of governors on cars driven by its sales staff. These devices would limit the car speeds to a preset level, which is expected to improve fuel economy. The company is planning to test several cars for fuel consumption without governors for 1 week. Then governors would be installed in the same cars, and fuel consumption will be monitored for another week. Gamma Corporation wants to estimate the mean difference in fuel consumption with a margin of error of estimate of \(2 \mathrm{mpg}\) with a \(90 \%\) confidence level. Assume that the differences in fuel consumption are normally distributed and that previous studies suggest that an estimate of \(s_{d}=3 \mathrm{mpg}\) is reasonable. How many cars should be tested? (Note that the critical value of \(t\) will depend on \(n\), so it will be necessary to use trial and error.)
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