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Chapter 7: Problem 17

Costs for a 30 -Second Spot on Cable Television The approximate costs for 30 -second randomly selected spots for various cable networks in a random selection of cities are shown. Estimate the true population mean cost for a 30 -second advertisement on cable network with \(90 \%\) confidence. $$ \begin{array}{lllllllll}{14} & {55} & {165} & {9} & {15} & {66} & {23} & {30} & {150} \\ {22} & {12} & {13} & {54} & {73} & {55} & {41} & {78}\end{array} $$

Short Answer

Expert verified
The 90% confidence interval for the mean cost is \([32.18, 70.76]\).

Step by step solution

01

Organize the Data

List all the given data points for the costs of a 30-second TV spot. These are: \(14, 55, 165, 9, 15, 66, 23, 30, 150, 22, 12, 13, 54, 73, 55, 41, 78\).
02

Calculate the Sample Mean

To find the sample mean \(\bar{x}\), add all the data points together and divide by the number of data points. \(\bar{x} = \frac{14 + 55 + 165 + 9 + 15 + 66 + 23 + 30 + 150 + 22 + 12 + 13 + 54 + 73 + 55 + 41 + 78}{17} = 51.47\).
03

Calculate the Sample Standard Deviation

First, find the differences between each data point and the mean, square them, sum these squared differences, and then divide by the number of data points minus one. Finally, take the square root of this value. \[ s = \sqrt{\frac{\sum{(x_i - \bar{x})^2}}{n-1}} = \sqrt{\frac{54065.06}{16}} = 45.60\]
04

Determine the Critical Value for 90% Confidence

For a 90% confidence level and a sample size of 17, find the critical value \(t^*\) using a t-distribution table. With \(n-1 = 16\) degrees of freedom, \(t^* \approx 1.746\).
05

Calculate the Margin of Error

The margin of error \(E\) is calculated using the formula: \[ E = t^* \times \frac{s}{\sqrt{n}} = 1.746 \times \frac{45.60}{\sqrt{17}} = 19.29 \].
06

Establish the Confidence Interval

The 90% confidence interval for the true mean cost is found by adding and subtracting the margin of error from the sample mean: \[ \bar{x} \pm E = 51.47 \pm 19.29 \]Thus, the interval is \([32.18, 70.76]\).

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Key Concepts

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

Sample Mean Calculation
The sample mean is like the average of a list of numbers and is crucial when estimating the population mean. It provides a single value that describes the central point of your dataset. To find it, you simply add up all your data points and divide by the total number of points you have. For instance:
\[ \bar{x} = \frac{14 + 55 + 165 + 9 + 15 + 66 + 23 + 30 + 150 + 22 + 12 + 13 + 54 + 73 + 55 + 41 + 78}{17} = 51.47 \]
This formula helps capture the average cost across different 30-second TV spots measured.
Sample Standard Deviation
The sample standard deviation tells us how spread out the data is around the mean. A larger standard deviation indicates more variability among the data points.
  • Start by finding the difference between each data point and the mean.
  • Square these differences to make them positive and then sum them up.
  • Divide by the number of observations minus one. This approach, known as Bessel's correction, provides an unbiased estimate of the population standard deviation.
  • Finally, take the square root to obtain the standard deviation.
For our sample data, it's calculated as:
\[ s = \sqrt{\frac{\sum{(x_i - \bar{x})^2}}{n-1}} = \sqrt{\frac{54065.06}{16}} = 45.60 \]
This value shows the typical deviation from the mean in the cost of TV spots.
Margin of Error
The margin of error helps understand how much we expect our sample mean to differ from the true population mean. It provides a range within which the population mean is likely to fall.
The margin of error is calculated with this formula:
\[ E = t^* \times \frac{s}{\sqrt{n}} \]
Where:
  • \( t^* \) is the critical value from the t-distribution.
  • \( s \) is the sample standard deviation.
  • \( n \) is the number of data points in the sample.
For our data, this calculates to:
\[ E = 1.746 \times \frac{45.60}{\sqrt{17}} = 19.29 \]
This means that our estimate of the mean cost could vary by this margin on either side.
t-Distribution Critical Value
The t-distribution critical value, denoted as \( t^* \), is part of finding the confidence interval. It accounts for uncertainties in small sample sizes. By referring to a t-distribution table, you find the critical value compatible with your desired level of confidence and sample size (or degrees of freedom, \( n-1 \)).
For a confidence level of 90% and a sample size of 17:
  • The degrees of freedom \( n-1 = 16 \).
  • We lookup \( t^* \) in a t-table, yielding \( t^* \approx 1.746 \).
This critical value scales the margin of error, ensuring that our interval estimate effectively captures the true mean cost of TV spots.

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Most popular questions from this chapter

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