n a survey, 44 people were asked how much they spent on their child's last birthday gift. The results were oughly bell-shaped with a mean of $70 and standard deviation of $6. Calculate, state, and interpret a 95% confidence interval to estimate the mean amount of money parents spend on their child's birthday gift. Round to the nearest 100th where necessary. Use the space below to type your In Or
n a survey, 44 people were asked how much they spent on their child's last birthday gift. The results were oughly bell-shaped with a mean of $70 and standard deviation of $6. Calculate, state, and interpret a 95% confidence interval to estimate the mean amount of money parents spend on their child's birthday gift. Round to the nearest 100th where necessary. Use the space below to type your In Or
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![### Understanding Confidence Intervals for Spending on Children's Birthday Gifts
In a survey, 44 people were asked how much they spent on their child's last birthday gift. The results were roughly bell-shaped with a mean of $70 and a standard deviation of $6.
Calculate, state, and interpret a 95% confidence interval to estimate the mean amount of money parents spend on their child's birthday gift. Round to the nearest 100th where necessary.
### Solution
To calculate the 95% confidence interval for the mean amount of money parents spend on their child's birthday gift, follow these steps:
1. **Identify the sample mean (\(\bar{x}\)) and sample standard deviation (s)**:
- Mean (\(\bar{x}\)) = $70
- Standard deviation (s) = $6
2. **Identify the sample size (n)**:
- Sample size (n) = 44
3. **Determine the critical value (z) for a 95% confidence interval**:
- For a 95% confidence level, the critical value (z) approximately equals 1.96.
4. **Calculate the standard error of the mean (SEM)**:
\[
SEM = \frac{s}{\sqrt{n}} = \frac{6}{\sqrt{44}} ≈ 0.905
\]
5. **Compute the margin of error (ME)**:
\[
ME = z \times SEM = 1.96 \times 0.905 ≈ 1.7748
\]
6. **Determine the confidence interval**:
\[
\text{Lower limit} = \bar{x} - ME ≈ 70 - 1.7748 ≈ 68.23
\]
\[
\text{Upper limit} = \bar{x} + ME ≈ 70 + 1.7748 ≈ 71.77
\]
Hence, the 95% confidence interval for the mean amount of money parents spend on their child's birthday gift is approximately \$68.23 to \$71.77.
Feel free to use the space below to type your answers and any additional calculations or interpretations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F44029bcf-9042-4df6-b542-d8d45dd97a34%2Fe304dfb5-c3d7-4a6f-8e1e-462807bdb865%2Fvjv52xj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Confidence Intervals for Spending on Children's Birthday Gifts
In a survey, 44 people were asked how much they spent on their child's last birthday gift. The results were roughly bell-shaped with a mean of $70 and a standard deviation of $6.
Calculate, state, and interpret a 95% confidence interval to estimate the mean amount of money parents spend on their child's birthday gift. Round to the nearest 100th where necessary.
### Solution
To calculate the 95% confidence interval for the mean amount of money parents spend on their child's birthday gift, follow these steps:
1. **Identify the sample mean (\(\bar{x}\)) and sample standard deviation (s)**:
- Mean (\(\bar{x}\)) = $70
- Standard deviation (s) = $6
2. **Identify the sample size (n)**:
- Sample size (n) = 44
3. **Determine the critical value (z) for a 95% confidence interval**:
- For a 95% confidence level, the critical value (z) approximately equals 1.96.
4. **Calculate the standard error of the mean (SEM)**:
\[
SEM = \frac{s}{\sqrt{n}} = \frac{6}{\sqrt{44}} ≈ 0.905
\]
5. **Compute the margin of error (ME)**:
\[
ME = z \times SEM = 1.96 \times 0.905 ≈ 1.7748
\]
6. **Determine the confidence interval**:
\[
\text{Lower limit} = \bar{x} - ME ≈ 70 - 1.7748 ≈ 68.23
\]
\[
\text{Upper limit} = \bar{x} + ME ≈ 70 + 1.7748 ≈ 71.77
\]
Hence, the 95% confidence interval for the mean amount of money parents spend on their child's birthday gift is approximately \$68.23 to \$71.77.
Feel free to use the space below to type your answers and any additional calculations or interpretations.
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