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Chapter 9: Problem 58

As part of an annual review of its accounts, a discount brokerage selects a random sample of 36 customers. Their accounts are reviewed for total account valuation, which showed a mean of \(\$ 32,000,\) with a sample standard deviation of \(\$ 8,200 .\) What is a 90 percent confidence interval for the mean account valuation of the population of customers?

Short Answer

Expert verified
The 90% confidence interval is \((29752.63, 34247.37)\).

Step by step solution

01

Identify the Sample Mean and Standard Deviation

From the problem, we identify that the sample mean \( \bar{x} \) is \( \\( 32,000 \), and the sample standard deviation \( s \) is \( \\) 8,200 \). The sample size \( n \) is 36.
02

Determine the Z-score for 90% Confidence

For a 90% confidence interval, we use a Z-score associated with the 90% level of confidence. Looking up a Z-table or using a calculator, the Z-score is approximately 1.645.
03

Calculate the Standard Error

The standard error (SE) of the mean is calculated using the formula \( SE = \frac{s}{\sqrt{n}} \). Substituting the values, we get \( SE = \frac{8200}{\sqrt{36}} = \frac{8200}{6} = 1366.67 \).
04

Calculate the Margin of Error

The margin of error (ME) is calculated using the formula \( ME = Z \times SE \). Substituting the values, \( ME = 1.645 \times 1366.67 \approx 2247.37 \).
05

Establish the Confidence Interval

The confidence interval is calculated using the formula \( \bar{x} \pm ME \). Substituting the values, the confidence interval is \( 32000 \pm 2247.37 \), which results in the interval \( (29752.63, 34247.37) \).

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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 key concept in statistics that involves calculating the average of a set of values obtained from a sample of a population. In simpler terms, it's the total sum of all observations in the sample divided by the number of observations. For instance, when reviewing the account valuation of a random sample of brokerage customers, we found that the sample mean was $32,000.
This number tells us the central value around which the account valuations of the sample cluster. Importantly, the sample mean is used as an estimate of the population mean, under the assumption that our sample is representative of the entire population.
Standard Deviation
Standard deviation is a measure of how much the values in a data set vary, or deviate, from the mean. A smaller standard deviation indicates that the values are clustered closer to the mean, while a larger standard deviation indicates that they are more spread out.
In our brokerage example, the sample standard deviation was $8,200. This means that the account valuations typically vary by about $8,200 from the sample mean of $32,000. Understanding standard deviation helps us grasp the data spread, which is crucial for making accurate forecasts and assessments.
Z-score
The Z-score is a statistical measure that describes a value's position in relation to the mean of a group of values, expressed in terms of standard deviations. It indicates how many standard deviations an element is from the mean.
In the calculation of a confidence interval, which aims to estimate the range within which a population mean lies, a Z-score is used to express the level of confidence. For a 90% confidence interval, as in our task, the Z-score is approximately 1.645. This Z-score tells us that 90% of the data should fall within this interval when the distribution is normal.
Standard Error
The standard error of the mean quantifies the variability of the sample mean compared to the population mean. It's calculated by dividing the standard deviation by the square root of the sample size.
In our scenario, the standard error was calculated to be approximately $1,366.67. A smaller standard error indicates that the sample mean provides a more accurate estimate of the population mean. This is important for creating reliable confidence intervals, as a lower standard error means that the estimated range (interval) for the mean is more precise.

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Most popular questions from this chapter

You plan to conduct a survey to find what proportion of the workforce has two or more jobs. You decide on the 95 percent confidence level and state that the estimated proportion must be within 2 percent of the population proportion. A pilot survey reveals that 5 of the 50 sampled hold two or more jobs. How many in the workforce should be interviewed to meet your requirements?

A survey is being planned to determine the mean amount of time corporation executives watch television. A pilot survey indicated that the mean time per week is 12 hours, with a standard deviation of 3 hours. It is desired to estimate the mean viewing time within one-quarter hour. The 95 percent level of confidence is to be used. How many executives should be surveyed?

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.

The Human Relations Department of Electronics, Inc., would like to include a dental plan as part of the benefits package. The question is: How much does a typical employee and his or her family spend per year on dental expenses? A sample of 45 employees reveals the mean amount spent last year was \(\$ 1,820,\) with a standard deviation of \(\$ 660 .\) a. Construct a 95 percent confidence interval for the population mean. b. The information from part (a) was given to the president of Electronics, Inc. He indicated he could afford \(\$ 1,700\) of dental expenses per employee. Is it possible that the population mean could be \(\$ 1,700 ?\) Justify your answer.

The online edition of the Information Please Almanac is a valuable source of business information. Go to the Website at www.infoplease.com. Click on Business. Then in the Almanac Section, click on Taxes, then click on State Taxes on Individuals. The result is a listing of the 50 states and the District of Columbia. Use a table of random numbers to randomly select 5 to 10 states. Compute the mean state tax rate on individuals. Develop a confidence interval for the mean amount. Because the sample is a large part of the population, you will want to include the finite population correction factor. Interpret your result. You might, as an additional exercise, download all the information and use Excel or MINITAB to compute the population mean. Compare that value with the results of your confidence interval.
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