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Chapter 9: Problem 31

A random sample of 85 group leaders, supervisors, and similar personnel at General Motors revealed that, on the average, they spent 6.5 years on the job before being promoted. The standard deviation of the sample was 1.7 years. Construct a 95 percent confidence interval.

Short Answer

Expert verified
The 95% confidence interval is [6.1388, 6.8612] years.

Step by step solution

01

Identify the Components of the Confidence Interval Formula

To calculate a confidence interval for the population mean, we can use the formula: \[\text{Confidence Interval} = \bar{x} \pm Z \left(\frac{\sigma}{\sqrt{n}}\right)\]where \(\bar{x}\) is the sample mean, \(Z\) is the Z-value that corresponds to the desired confidence level, \(\sigma\) is the standard deviation, and \(n\) is the sample size. For this problem: \(\bar{x} = 6.5\), \(\sigma = 1.7\), and \(n = 85\).
02

Determine the Z-value for 95% Confidence Interval

For a 95% confidence interval, the Z-value (Z-score) is typically found using the standard normal distribution table. The Z-value corresponding to a 95% confidence level is approximately 1.96.
03

Calculate the Standard Error

The standard error of the mean (SE) is found by the formula:\[SE = \frac{\sigma}{\sqrt{n}} = \frac{1.7}{\sqrt{85}}\]Calculating the value, we find:\[SE \approx \frac{1.7}{9.22} \approx 0.1843\]
04

Calculate the Confidence Interval

Use the formula for the confidence interval:\[\text{Confidence Interval} = \bar{x} \pm Z \times SE\]Substitute the values:\[6.5 \pm 1.96 \times 0.1843\]This results in:\[6.5 \pm 0.3612\]Thus, the confidence interval is \([6.1388, 6.8612]\).
05

Interpret the Results

We interpret this confidence interval as: We are 95% confident that the true average number of years employees like group leaders and supervisors at General Motors spent on the job before being promoted is between 6.1388 and 6.8612 years.

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Key Concepts

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

Sample Mean
The sample mean is a critical concept, especially when estimating population parameters. It represents the average value calculated from a sample, which is a smaller group chosen from a larger population. In our exercise, the sample mean (\(\bar{x}\)) is 6.5 years. This means that the sampled group leaders, supervisors, and similar personnel at General Motors spent, on average, 6.5 years on the job before being promoted.
  • It serves as an estimate of the population mean.
  • It is essential for constructing confidence intervals.
  • As the sample size increases, the sample mean becomes a more accurate estimate of the population mean.
Standard Error
The standard error (SE) measures how much variability or dispersion we can expect in a sample mean. Think of it as a way to understand how the sample mean might differ from the true population mean. In this exercise, the standard error was calculated using the formula:\[SE = \frac{\sigma}{\sqrt{n}} = \frac{1.7}{\sqrt{85}}\]This resulted in a standard error of approximately 0.1843 years.
  • It depends on both the sample size and the standard deviation of the sample.
  • A larger sample size will usually result in a smaller standard error, indicating more precise estimates of the population mean.
  • Standard error is utilized when calculating confidence intervals, allowing us to estimate the range in which the population mean likely falls.
Z-score
A Z-score is a statistical measurement that describes a value's relation to the mean of a group of values. In confidence interval calculations, the Z-score represents the number of standard deviations a data point is from the mean. For the 95% confidence interval in our example, the Z-score is 1.96. This value is typically found from the standard normal distribution table.
  • The Z-score determines how "extreme" a value is compared to the average.
  • When calculating a 95% confidence interval, this Z-score ensures that the interval will contain the population mean 95% of the time.
  • Different confidence levels will correspond to different Z-scores.
Population Mean
The population mean is the average of all potential observations in a population. It is what researchers ultimately aim to estimate through their sample data. In the context of our exercise, it refers to the average number of years all group leaders, supervisors, and similar positions at General Motors spend on the job before being promoted.
To estimate the population mean, we use tools like confidence intervals:
  • Confidence intervals give a range within which the true population mean is likely to fall.
  • In our exercise, the calculated confidence interval was between 6.1388 and 6.8612 years, meaning the true population mean is likely in that range.
  • The accuracy and reliability of the population mean estimation improve with increased sample sizes and reduced standard error.

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Most popular questions from this chapter

A sample of 10 observations is selected from a normal population for which the population standard deviation is known to be 5 . The sample mean is 20 . a. Determine the standard error of the mean. b. Explain why we can use formula \((9-1)\) to determine the 95 percent confidence interval even though the sample is less than \(30 .\) c. Determine the 95 percent confidence interval for the population mean.

14\. The Greater Pittsburgh Area Chamber of Commerce wants to estimate the mean time workers who are employed in the downtown area spend getting to work. A sample of 15 workers reveals the following number of minutes spent traveling. \begin{tabular}{|llllllll|} \hline 29 & 38 & 38 & 33 & 38 & 21 & 45 & 34 \\ 40 & 37 & 37 & 42 & 30 & 29 & 35 & \\ \hline \end{tabular} Develop a 98 percent confidence interval for the population mean. Interpret the result.

A processor of carrots cuts the green top off each carrot, washes the carrots, and inserts six to a package. Twenty packages are inserted in a box for shipment. To test the weight of the boxes, a few were checked. The mean weight was 20.4 pounds, the standard deviation 0.5 pounds. How many boxes must the processor sample to be 95 percent confident that the sample mean does not differ from the population mean by more than 0.2 pound?

The attendance at the Savannah Colts minor league baseball game last night was \(400 .\) A random sample of 50 of those in attendance revealed that the mean number of soft drinks consumed per person was 1.86 with a standard deviation of \(0.50 .\) Develop a 99 percent confidence interval for the mean number of soft drinks consumed per person.

There are 300 welders employed at Maine Shipyards Corporation. A sample of 30 welders revealed that 18 graduated from a registered welding course. Construct the 95 percent confidence interval for the proportion of all welders who graduated from a registered welding course.
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