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Chapter 8: Problem 45

activities (playing games, personal communications, etc.) during this month are as follows: $$ \begin{array}{lllllllll} 7 & 12 & 9 & 8 & 11 & 4 & 14 & 1 & 6 \end{array} $$ Assuming that such times for all employees are approximately normally distributed, make a \(95 \%\) confidence interval for the corresponding population mean for all employees of this company.A company randomly selected nine office employees and secretly monitored their computers for one month. The times (in hours) spent by these employees using their computers for non- job-related

Short Answer

Expert verified
The 95% confidence interval for the population mean of all employees of this company is the sample mean plus/minus the margin of error. Use your calculations from steps 1-4 to find these values.

Step by step solution

01

Calculate the sample mean

The first thing to do is to calculate the sample mean. This can be done by adding all the values together and then dividing by the number of values. Here, exchange the values \(7, 12, 9, 8, 11, 4, 14, 1, 6\) and divide by 9.
02

Calculate the sample standard deviation

Next, calculate the sample standard deviation. Subtract the mean from each value, square the results, sum them, divide by n-1 (where n is the sample size), and take the square root. This will give the standard deviation.
03

Determine the z-score for the given confidence level

Look up the standard z-value that corresponds to a 95% confidence level. This is usually found in a table or calculator. Because we want a range and not just a single direction, we have to split the 5% error between two directions, so we鈥檒l use the z-value that corresponds to 97.5% (or 0.975), which is 1.96.
04

Calculate the margin of error

The margin of error can be calculated by multiplying the z-score by the standard error. The standard error is the standard deviation divided by the square root of the sample size.
05

Calculate the confidence interval

The confidence interval is found by taking the sample mean and adding and subtracting the margin of error.

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Key Concepts

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

Sample Mean
When dealing with statistical samples, the **sample mean** is a central concept. It represents the average of all the data points in your sample. Calculating it is simple and intuitive鈥攋ust sum up all the data values and divide by the number of data points.
For example, given the data set \([7, 12, 9, 8, 11, 4, 14, 1, 6]\), the sum is 72. Dividing 72 by the number of values, which is 9, gives a sample mean of 8.
  • The sample mean provides a quick glimpse into the central tendency of your data set.
  • It serves as an estimate of the population mean.
  • To remember, it's simply the arithmetic mean of your sample values.
Understanding the sample mean is essential as it's the starting point for forming a confidence interval.
Sample Standard Deviation
The **sample standard deviation** measures how spread out the numbers are in your data set. Understanding this helps us grasp how much the individual data points vary from the sample mean. To calculate it accurately, follow these steps:
  • Subtract the sample mean from each data point.
  • Square each result to eliminate negative values.
  • Sum these squares.
  • Divide this sum by the number of data points minus one (n-1).
  • Lastly, take the square root of this quotient.

For instance, from our earlier data, after following these steps, you would find the standard deviation. This number is crucial because it gives you the standard error in the confidence interval calculation. Seeing how tightly or loosely data points cluster around the mean can inform a lot about their reliability.
Z-Score
The **z-score** is used in statistics to understand how one data point compares to the typical data point in a data set. For confidence intervals, the z-score helps determine how far your sample mean is from the true population mean.
To find the z-score needed for a 95% confidence interval, consult a statistical table or calculator. For instance, to make a two-tailed 95% interval, you look for a z-score that corresponds to 97.5%. This is because you allocate 0.025 to each tail of the distribution.
  • A commonly used z-score for 95% confidence is 1.96.
  • This z-score reflects the number of standard deviations your sample's mean is from the population mean.
  • It's critical for constructing the margin of error.

By incorporating the z-score, you can develop reliable predictions about the population based on your sample.
Margin of Error
The **margin of error** indicates the range of uncertainty around your sample mean estimate. It's a central part of any confidence interval and helps quantify how exact you expect your average to be.
To calculate the margin of error, you multiply the z-score by the **standard error** of your sample mean (standard deviation divided by the square root of n).
  • This represents how much your sample mean could vary from the true population mean.
  • It assists in constructing upper and lower bounds for your confidence interval.
  • The smaller the margin of error, the more precise your sample mean estimate is.
Ultimately, the margin of error adds context to your confidence interval, showing the possible range of values for the population mean.

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Most popular questions from this chapter

For a population data set, \(\sigma=14.50\). a. What should the sample size be for a \(98 \%\) confidence interval for \(\mu\) to have a margin of error of estimate equal to \(5.50\) ? b. What should the sample size be for a \(95 \%\) confidence interval for \(\mu\) to have a margin of error of estimate equal to \(4.25\) ?

York Steel Corporation produces iron rings that are supplied to other companies. These rings are supposed to have a diameter of 24 inches. The machine that makes these rings does not produce each ring with a diameter of exactly 24 inches. The diameter of each of the rings varies slightly. It is known that when the machine is working properly, the rings made on this machine have a mean diameter of 24 inches. The standard deviation of the diameters of all rings produced on this machine is always equal to \(.06\) inch. The quality control department takes a random sample of 25 such rings every week, calculates the mean of the diameters for these rings, and makes a \(99 \%\) confidence interval for the population mean. If either the lower limit of this confidence interval is less than \(23.975\) inches or the upper limit of this confidence interval is greater than \(24.025\) inches, the machine is stopped and adjusted. A recent such sample of 25 rings produced a mean diameter of \(24.015\) inches. Based on this sample, can you conclude that the machine needs an adjustment? Explain. Assume that the population distribution is approximately normal.

A bank manager wants to know the mean amount of mortgage paid per month by homeowners in an area. A random sample of 120 homeowners selected from this area showed that they pay an average of \(\$ 1575\) per month for their mortgages. The population standard deviation of all such mortgages is \(\$ 215\). a. Find a \(97 \%\) confidence interval for the mean amount of mortgage paid per month by all homeowners in this area. b. Suppose the confidence interval obtained in part a is too wide. How can the width of this interval be reduced? Discuss all possible alternatives. Which alternative is the best?

A researcher wants to determine a \(99 \%\) confidence interval for the mean number of hours that adults spend per week doing community service. How large a sample should the researcher select so that the estimate is within \(1.2\) hours of the population mean? Assume that the standard deviation for time spent per week doing community service by all adults is 3 hours.

The standard deviation for a population is \(\sigma=14.8\). A random sample of 25 observations selected from this population gave a mean equal to \(143.72\). The population is known to have a normal distribution. a. Make a \(99 \%\) confidence interval for \(\mu\). b. Construct a \(95 \%\) confidence interval for \(\mu\). c. Determine a \(90 \%\) confidence interval for \(\mu\). d. Does the width of the confidence intervals constructed in parts a through \(\mathrm{c}\) decrease as the confidence level decreases? Explain your answer.
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