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Chapter 9: Problem 51

Suppose that 200 statistics students each took a random sample (with replacement) of 50 students at their college and recorded the ages of the students in their sample. Then each student used his or her data to calculate a \(95 \%\) confidence interval for the mean age of all students at the college. How many of the 200 intervals would you expect to capture the true population mean age, and how many would you expect not to capture the true population mean? Explain by showing your calculation.

Short Answer

Expert verified
You would expect about 190 of the 200 confidence intervals to capture the true population mean age, and about 10 of them not to capture the true population mean age.

Step by step solution

01

Understanding the concept of confidence intervals

The concept of a confidence interval is that it provides a range within which the true population parameter lies with a certain level of confidence. Here, a 95% confidence level is provided, meaning that we are 95% confident that the true population parameter lies within the interval.
02

Apply the concept to the problem

In this case, each student has calculated a 95% confidence interval for the mean age. Therefore, for every 100 intervals calculated, we can expect 95 of them to capture the true population mean.
03

Calculation

Since there are 200 students, we multiply the total number of intervals(200) by the confidence level (0.95) to get the expected number of intervals that capture the true mean. Likewise, to know how many intervals we expect not to capture the true population mean, we multiply the number of intervals(200) by (1 - confidence level) = (1 - 0.95)
04

Result

By calculation, 0.95 * 200 = 190, that is, expect about 190 intervals to capture the true population mean. And, 0.05 * 200 =10, expect approximately 10 intervals not to capture the true population mean.

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Key Concepts

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

Population Mean
The population mean is a fundamental concept in statistics. It refers to the average value of a particular characteristic across a whole population. For example, if you want to know the average age of all students at a college, the true value is the population mean.
  • In many practical situations, it is challenging to collect data from an entire population. For this reason, researchers often rely on samples to estimate the population mean.
  • The estimated mean from the sample is used to infer the population mean with a degree of confidence provided by the confidence interval.
By understanding the population mean, we can make more informed decisions and predictions based on the data we have.
Sample Size
Sample size is the number of observations or data points collected from a population in a study. It plays a critical role in the accuracy of statistical estimates such as the confidence interval.
  • A larger sample size tends to give more reliable estimates of the population parameters because it better represents the population.
  • Smaller sample sizes may result in wider confidence intervals, indicating more uncertainty about the estimate of the population mean.
In our earlier example, each student took a random sample of 50 students from the college to estimate the mean age. This sample size, combined with calculating confidence intervals, helps determine how accurately the sample reflects the population.
Confidence Level
The confidence level in statistics expresses how certain we are that the true parameter lies within the calculated interval. A common practice is to use a confidence level of 95%, which means we expect that 95 out of 100 calculated confidence intervals will contain the true population parameter.
  • The confidence level influences the width of the confidence interval. Higher confidence levels result in wider intervals; this is a trade-off for having greater certainty.
  • In practical terms, a 95% confidence level in our problem implies that we are 95% sure about capturing the true mean age of college students in our confidence interval.
This concept is vital because it quantifies the uncertainty associated with sample estimates and is key to understanding the reliability of statistical conclusions.

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Most popular questions from this chapter

Use the data from exercise \(9.36\). a. Using the four-step procedure with a two-sided alternative hypothesis, should you be able to reject the hypothesis that the population mean is 5 pounds using a significance level of \(0.05\) ? Why or why not? The confidence interval is reported here: I am \(95 \%\) confident the population mean is between \(4.9\) and \(5.3\) pounds. b. Now test the hypothesis that the population mean is not 5 pounds using the four-step procedure. Use a significance level of \(0.05\) and number your steps.

According to a 2017 report by ComScore .com, the mean time spent on smartphones daily by the American adults is \(2.85\) hours. Assume this is correct and assume the standard deviation is \(1.4\) hours. a. Suppose 150 American adults are randomly surveyed and asked how long they spend on their smartphones daily. The mean of the sample is recorded. Then we repeat this process, taking 1000 surveys of 150 American adults and recording the sample means. What will be the shape of the distribution of these sample means? b. Refer to part (a). What will be the mean and the standard deviation of the distribution of these sample means?

State whether each situation has independent or paired (dependent) samples. a. A researcher wants to compare food prices at two grocery stores. She purchases 20 items at Store \(A\) and finds the mean and the standard deviation for the cost of the items. She then purchases 20 items at Store \(\mathrm{B}\) and again finds the mean and the standard deviation for the cost of the items. b. A student wants to compare textbook prices at two bookstores. She has a list of textbooks and finds the price of the text at each of the two bookstores.

The distribution of the scores on a certain exam is \(N(100,10)\) which means that the exam scores are Normally distributed with a mean of 100 and a standard deviation of \(10 .\) a. Sketch or use technology to create the curve and label on the \(x\) -axis the position of the mean, the mean plus or minus one standard deviation, the mean plus or minus two standard deviations, and the mean plus or minus three standard deviations. b. Find the probability that a randomly selected score is between 90 and \(110 .\) Shade the region under the Normal curve whose area corresponds to this probability.

The weights of four randomly chosen bags of horse carrots, each bag labeled 20 pounds, were \(20.5,19.8,20.8\), and \(20.0\) pounds. Assume that the distribution of weights is Normal. Find a \(95 \%\) confidence interval for the mean weight of all bags of horse carrots. Use technology for your calculations. a. Decide whether each of the following three statements is a correctly worded interpretation of the confidence interval, and fill in the blanks for the correct option(s). i. \(95 \%\) of all sample means based on samples of the same size will be between _____ and _____. ii. I am \(95 \%\) confident that the population mean is between _____ and _____. iii. We are \(95 \%\) confident that the boundaries are _____ and _____. b. Can you reject a population mean of 20 pounds? Explain.
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