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Chapter 8: Problem 16

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 \(4.25\) ?

Short Answer

Expert verified
So, the suitable sample sizes for a 98% and 95% confidence intervals having margins of error of 5.50 and 4.25 respectively are 51 and 98.

Step by step solution

01

- Finding Z score for 98% confidence interval

The first step is to find the z-score corresponding to the given level of confidence, which is 98%. Usually, the Z value for a 98 % confidence interval are obtained from Standard Normal Distribution Table which is approximately \(2.33\).
02

- Calculate the sample size for 98% confidence interval

Using the formula mentioned in the analysis and applying the known values to it \(n = \left(\frac{2.33 * 14.5}{5.50}\right)^2\). Solving this gives us approximately \(50.04\), which then needs to be rounded up to the nearest whole number since it's not possible to have fractional sample size. Thus, the sample size should be \(51\).
03

- Finding Z score for 95% confidence interval

After solving for 98%, we will now find the Z score for the 95% confidence interval. Usually, the Z value for a 95% confidence interval obtained from Standard Normal Distribution Table is approximately \(1.96\).
04

- Calculate the sample size for 95% confidence interval

Substitute the known values into the formula \(n = \left(\frac{1.96 * 14.5}{4.25}\right)^2\). Solving this we get approximately \(97.79\), which then needs to be rounded up to the nearest whole number. Thus, the sample size should be \(98\).

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Key Concepts

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

Understanding the Z-Score
The z-score is a statistical measure that expresses the number of standard deviations a value, or data point, is away from the mean of a data set. It is a crucial component in calculating confidence intervals which help us understand the range within which we expect a population parameter, like the mean, to fall.
To find the z-score for a specific confidence level, statisticians often refer to a standard normal distribution table. For instance:
  • A 98% confidence interval typically corresponds to a z-score of approximately 2.33.
  • A 95% confidence interval corresponds to a z-score of around 1.96.
These z-scores reflect the desired degree of certainty that the parameter lies within the interval. The larger the z-score, the wider the interval, signifying greater confidence in capturing the parameter.
Decoding the Margin of Error
The margin of error represents the range of uncertainty in an estimate. It specifies how much the sample mean might differ from the actual population mean. When constructing a confidence interval, the margin of error is added to and subtracted from the sample mean to create the interval range.
A smaller margin of error requires a larger sample size, as it indicates you want your estimate to closely represent the population mean with fewer fluctuations. For instance:
  • A margin of error of 5.50 (in a 98% confidence interval) and 4.25 (in a 95% confidence interval) necessitate specific sample sizes calculated using the formula: \[ n = \left(\frac{z \times \sigma}{E}\right)^2 \]
Where \(z\) is the z-score, \(\sigma\) is the standard deviation, and \(E\) is the margin of error.
Unpacking Standard Deviation
Standard deviation is a measure that indicates the amount of variation or dispersion in a set of values. It tells us how tightly the data points are clustered around the mean. A lower standard deviation means that data points tend to be close to the mean, while a higher standard deviation indicates a wider spread.
  • In the exercise at hand, a standard deviation of 14.50 is given. This number is key because it influences the confidence interval width and required sample size.
  • A large standard deviation often translates to larger confidence intervals for the same margin of error, requiring a larger sample size to maintain precision.
Understanding standard deviation is crucial because it directly impacts statistical calculations like those needed for confidence intervals.
Exploring the Population Data Set
A population data set includes all the elements from a defined group that one wants to study. For example, if you are studying the heights of high school students in a city, every high school student's height in that city would be part of the population data set.
  • The goal of statistical analysis with such data sets is often to make inferences about the population from a smaller sample.
  • In calculating sample size, the overall variability within the population (represented by standard deviation) and the desired margin of error play significant roles.
By sampling effectively from the population data set and calculating the necessary sample size, researchers can achieve precise estimates with a known level of confidence.

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Most popular questions from this chapter

You are working for a bank. The bank manager wants to know the mean waiting time for all customers who visit this bank. She has asked you to estimate this mean by taking a sample. Briefly explain how you will conduct this study. Collect data on the waiting times for 45 customers who visit a bank. Then estimate the population mean. Choose your own confidence level.

A gas station attendant would like to estimate \(p\), the proportion of all households that own more than two vehicles. To obtain an estimate, the attendant decides to ask the next 200 gasoline customers how many vehicles their households own. To obtain an estimate of \(p\), the attendant counts the number of customers who say there are more than two vehicles in their households and then divides this number by \(200 .\) How would you critique this estimation procedure? Is there anything wrong with this procedure that would result in sampling and/or nonsampling errors? If so, can you suggest a procedure that would reduce this error?

A consumer agency wants to estimate the proportion of all drivers who wear seat belts while driving. Assume that a preliminary study has shown that \(76 \%\) of drivers wear seat belts while driving. How large should the sample size be so that the \(99 \%\) confidence interval for the population proportion has a margin of error of \(.03\) ?

A random sample of 34 participants in a Zumba dance class had their heart rates measured before and after a moderate 10 -minute workout. The following data correspond to the increase in each individual's heart rate (in beats per minute): \(\begin{array}{llllllllllll}59 & 70 & 57 & 42 & 57 & 59 & 41 & 54 & 44 & 36 & 59 & 61 \\ 52 & 42 & 41 & 32 & 60 & 54 & 52 & 53 & 51 & 47 & 62 & 62 \\ 44 & 69 & 50 & 37 & 50 & 54 & 48 & 52 & 61 & 45 & & \end{array}\) a. What is the point estimate of the corresponding population mean? b. Make a \(98 \%\) confidence interval for the average increase in a person's heart rate after a moderate 10 -minute Zumba workout.

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, on average, 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: \(\begin{array}{rrrrrrrrr}16.96 & 38.83 & 15.28 & 14.84 & 5.99 & 64.50 & 12.15 & 14.68 & 33.37 \\ 37.10 & 18.15 & 67.89 & 12.17 & 40.13 & 5.51 & 8.80 & 34.53 & 35.54 \\ 8.51 & 37.18 & 41.52 & 13.83 & 12.96 & 22.78 & 5.29 & 9.09 & \end{array}\) 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.
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