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Chapter 8: Problem 13

The standard deviation for a population is \(\sigma=15.3\). A sample of 36 observations selected from this population gave a mean equal to \(74.8\). a. Make a \(90 \%\) confidence interval for \(\mu .\) b. Construct a \(95 \%\) confidence interval for \(\mu .\) c. Determine a \(99 \%\) confidence interval for \(\mu .\) d. Does the width of the confidence intervals constructed in parts a through \(\mathrm{c}\) increase as the confidence level increases? Explain your answer.

Short Answer

Expert verified
a) 90% confidence interval for 渭 = [71.61, 77.99], b) 95% confidence interval for 渭 = [69.81, 79.79], c) 99% confidence interval for 渭 = [67.57, 82.03], d) Yes, the width of the confidence intervals increases as the confidence level increases.

Step by step solution

01

Calculate 90% confidence interval

First, find the Z score for 90% confidence level, which is 1.645. Then, compute the standard error (SE) of the sample, which is \(\sigma/\sqrt{n}\). So, SE = 15.3/鈭�36 = 2.55. Finally, the 90% confidence interval for 渭 is \(X虅 卤 (Z * SE)\) = 74.8 卤 (1.645 * 2.55) = [71.61, 77.99] .
02

Construct 95% confidence interval

The Z score for 95% confidence level is 1.96. Using the same process as in step 1, the 95% confidence interval for 渭 is 74.8 卤 (1.96 * 2.55) = [69.81, 79.79].
03

Determine 99% confidence interval

At a 99% confidence level, the Z score is 2.58. So, the 99% confidence interval for 渭 is 74.8 卤 (2.58 * 2.55) = [67.57, 82.03].
04

Observe the width of intervals

The width of the confidence intervals calculated in steps 1-3 clearly shows an increase as the confidence level increases. This is because a higher confidence level means more certainty, and to have more certainty, the range has to be wider to capture the mean 渭.

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Key Concepts

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

Standard Deviation
Standard deviation is a measure of how spread out numbers are in a data set. It tells us how much the individual data points
  • deviate from the mean (average).
  • It is denoted by the Greek letter sigma (\( \sigma \)).
A smaller standard deviation means that the data points are closer to the mean, while a larger one indicates that they are spread out over a wider range of values.
In the original exercise, the population's standard deviation is given as 15.3. This statistical measure helps to quantify the expected variability or spread of the sample data chosen from the population. When calculating the confidence interval, it's important to use the standard deviation to estimate the standard error of the sample mean.
Sample Mean
The sample mean, represented by \( \bar{X}\), is the average value of a sample set of data from a larger population. It provides an estimate of the population mean (\( \mu \)). To calculate the sample mean, you add up all the values in the sample and divide by the number of observations in the sample.
  • Sample mean = (sum of all sample values) / number of sample values.
In the exercise, the sample mean is given as 74.8, which serves as an estimate of the population mean for purposes of constructing confidence intervals.
Using the sample mean helps us make predictions about the population mean when combined with standard deviation and the sample size.
Z Score
A Z score is a statistical measurement that describes a value's relationship to the mean of a group of values in terms of standard deviations. Essentially, it tells us how far and in what direction, in terms of standard deviation units, a value is from the mean.
  • A positive Z score indicates the value is above the mean.
  • A negative Z score shows it is below the mean.
The Z score is a key component in constructing a confidence interval. Depending on the desired confidence level, specific Z scores are used.
In the exercise, different Z scores were used for each confidence level:
  • 1.645 for 90%
  • 1.96 for 95%
  • 2.58 for 99%
These reflect the areas in the standard normal distribution for those levels of confidence.
Confidence Level
Confidence level indicates the percentage of times we'd expect the true population mean to fall within a calculated confidence interval if we were to repeat the sampling process numerous times. Higher confidence levels translate into wider confidence intervals, meaning more certainty that the true mean is captured within the interval, but they are less precise.
In the exercise, confidence levels of 90%, 95%, and 99% were demonstrated, each using a respective Z score to calculate the interval.
  • In a 90% confidence interval, there is a 90% chance that the interval encompasses the population mean.
  • Similarly for 95% and 99%, the intervals provide progressively higher certainty but at the cost of wider intervals.
Understanding how confidence levels affect the width of intervals helps us make more informed decisions about the reliability and precision needed in statistical estimations.

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Most popular questions from this chapter

A businesswoman is considering whether to open a coffee shop in a local shopping center. Before making this decision, she wants to know how much money people spend per week at coffee shops in that area. She took a random sample of 26 customers from the area who visit coffee shops and asked them to record the amount of money (in dollars) they would spend during the next week at coffee shops. At the end of the week, she obtained the following data (in dollars) from these 26 customers: \(\begin{array}{rrrrrrrr}16.96 & 38.83 & 15.28 & 14.84 & 5.99 & 64.50 & 12.15 & 14.68 & 33.37 \\ 37.10 & 18.15 & 67.89 & 12.17 & 40.13 & 5.51 & 8.80 & 34.53 & 35.54 \\ 8.51 & 37.18 & 41.52 & 13.83 & 12.96 & 22.78 & 5.29 & 9.09 & \end{array}\) Assume that the distribution of weekly expenditures at coffee shops by all customers who visit coffee shops in this area is approximately normal. a. What is the point estimate of the corresponding population mean? b. Make a \(95 \%\) confidence interval for the average amount of money spent per week at coffee shops by all customers who visit coffee shops in this area.

Refer to Exercise \(8.24\). A city planner wants to estimate, with a \(97 \%\) confidence level, the average monthly residential water usage in the city. Based on earlier data, the population standard deviation of the monthly residential water usage in this city is \(389.60\) gallons. How large a sample should be selected so that the estimate for the average monthly residential water usage in this city is within 100 gallons of the population mean?

Check if the sample size is large enough to use the normal distribution to make a confidence interval for \(p\) for each of the following cases. a. \(n=50\) and \(\hat{p}=.25\) b. \(n=160\) and \(\hat{p}=.03\) c. \(n=400\) and \(\hat{p}=.65\) d. \(n=75 \quad\) and \(\quad \hat{p}=.06\)

At Farmer's Dairy, a machine is set to fill 32 -ounce milk cartons. However, this machine does not put exactly 32 ounces of milk into each carton; the amount varies slightly from carton to carton. It is known that when the machine is working properly, the mean net weight of these cartons is 32 ounces. The standard deviation of the amounts of milk in all such cartons is always equal to \(.15\) ounce. The quality control department takes a sample of 25 such cartons every week, calculates the mean net weight of these cartons, and makes a \(99 \%\) confidence interval for the population mean. If either the upper limit of this confidence interval is greater than \(32.15\) ounces or the lower limit of this confidence interval is less than \(31.85\) ounces, the machine is stopped and adjusted. A recent sample of 25 such cartons produced a mean net weight of \(31.94\) ounces. Based on this sample, will you conclude that the machine needs an adjustment? Assume that the amounts of milk put in all such cartons have a normal distribution.

The mean time taken to design a house plan by 40 architects was found to be 23 hours with a standard deviation of \(3.75\) hours. a. Construct a \(98 \%\) confidence interval for the population mean \(\mu .\) b. Suppose the confidence interval obtained in part a is too wide. How can the width of this interval be reduced? Describe all possible alternatives. Which alternative is the best and why?
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