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Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter2: Equations And Inequalities

Section: Chapter Questions

Problem 59RE

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Question

### Testing a Claim about the Average Textbook Price

Suppose that BC's financial aid office allocates a textbook stipend by claiming that the average textbook at the BC bookstore costs $66.77. You want to test this claim.

#### Formulating Hypotheses
The null and alternative hypothesis in symbols would be:
- \( H_0 : \mu = 66.77 \)
- \( H_1 : \mu \neq 66.77 \)

(The fourth option from the top is correct in this case.)

#### Null Hypothesis in Words
The null hypothesis in words would be:
- The average price of all textbooks from the store is $66.77.

#### Data Observed
Based on a sample of 60 textbooks at the store, you find:
- Average price: $75.97
- Standard deviation: $25.4

#### Calculations
**Point Estimate**

The point estimate is the sample mean:

\[ 75.97 \]

**90% Confidence Interval**

To find the 90% confidence interval for the mean, use the formula for the confidence interval for a population mean (\(\mu\)) when the standard deviation (\(s\)) is known. The formula is:

\[
\text{CI} = \bar{x} \pm z^* \left( \frac{s}{\sqrt{n}} \right)
\]

Where:
- \( \bar{x} \) is the sample mean
- \( z^* \) is the z-value corresponding to the 90% confidence level (which is approximately 1.645 for a two-tailed test)
- \( s \) is the sample standard deviation
- \( n \) is the sample size

Substitute the values:
\[ \bar{x} = 75.97 \]
\[ s = 25.4 \]
\[ n = 60 \]
\[ z^* = 1.645 \]

\[
\text{CI} = 75.97 \pm 1.645 \left( \frac{25.4}{\sqrt{60}} \right)
\]

First, calculate the standard error:
\[
\frac{25.4}{\sqrt{60}} \approx 3.28
\]

Next, multiply by \( z^* \):
\[
1.645 \times 3.28 \approx 5.39
\]

So, the confidence

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