Questions: HW10: Problem 4 (10 points) Starting salaries of 64 college graduates who have taken a statistics course have a mean of 42,500 with a standard deviation of 6,800. Find an 90% confidence interval for μ. (NOTE: Do not use commas or dollar signs in your answers. Round each bound to three decimal places.) Lower-bound: Upper-bound: Note: You can earn partial credit on this problem.

HW10: Problem 4
(10 points)
Starting salaries of 64 college graduates who have taken a statistics course have a mean of 42,500 with a standard deviation of 6,800. Find an 90% confidence interval for μ. (NOTE: Do not use commas or dollar signs in your answers. Round each bound to three decimal places.)

Lower-bound: Upper-bound:

Note: You can earn partial credit on this problem.

Transcript text: HW10: Problem 4 (10 points) Starting salaries of 64 college graduates who have taken a statistics course have a mean of $\$ 42,500$ with a standard deviation of $\$ 6,800$. Find an $90 \%$ confidence interval for $\mu$. (NOTE: Do not use commas or dollar signs in your answers. Round each bound to three decimal places.) Lower-bound: $\square$ Upper-bound: $\square$ Note: You can earn partial credit on this problem. Preview My Answers Submit Answers You have attempted this problem 0 times. You have unlimited attempts remaining. Show Past Answers Page generated at 11/04/2024 at 05:21pm CST WeBWorK © 1996-2023 | theme: math4 | ww_version: 2.18 | pg_version 2.18 The WeBWorK Project

Solution

Solution Steps

Step 1: Given Information

We are given the following information about the starting salaries of 64 college graduates who have taken a statistics course:

Sample mean ($\bar{x}$): $42500$
Sample standard deviation ($s$): $6800$
Sample size ($n$): $64$
Confidence level: $90\%$

Step 2: Determine the Z-Score

For a $90\%$ confidence level, the significance level ($\alpha$) is: \[ \alpha = 1 - 0.90 = 0.10 \] Since we are looking for a two-tailed confidence interval, we divide $\alpha$ by $2$: \[ \frac{\alpha}{2} = 0.05 \] The corresponding Z-score for $0.05$ in the upper tail is approximately $1.645$.

Step 3: Calculate the Margin of Error

The margin of error ($E$) can be calculated using the formula: \[ E = z \cdot \frac{s}{\sqrt{n}} \] Substituting the values: \[ E = 1.645 \cdot \frac{6800}{\sqrt{64}} = 1.645 \cdot \frac{6800}{8} = 1.645 \cdot 850 = 1398.25 \]

Step 4: Calculate the Confidence Interval

The confidence interval is given by: \[ \bar{x} \pm E \] Calculating the lower and upper bounds: \[ \text{Lower bound} = 42500 - 1398.25 = 41101.75 \] \[ \text{Upper bound} = 42500 + 1398.25 = 43898.25 \]