A study of a local high school tried to determine the mean number of text messages that each student sent per day. The study surveyed a random sample of 112 students in the high school and found a mean of 191 messages sent per day with a standard deviation of 91 messages. At the 95% confidence level, find the margin of error for the mean, rounding to the nearest whole number. (Do not write plus or minus±).
Answer
Attempt 1 out of 2

To find the margin of error for the mean number of text messages sent per day, we can use the formula for the margin of error (ME) in the context of a confidence interval:

\[
\text{Margin of Error (ME)} = z \times \frac{s}{\sqrt{n}}
\]

Where:
- \( z \) is the z-score corresponding to the 95% confidence level.
- \( s \) is the sample standard deviation.
- \( n \) is the sample size.

Given:
- Sample size (\( n \)) = 112
- Sample mean (\( \bar{x} \)) = 191 (but we won't need this to calculate the margin of error)
- Sample standard deviation (\( s \)) = 91
- Confidence level = 95%

For a 95% confidence level, the corresponding z-score is approximately 1.96.

Now, let's calculate the margin of error:

\[
\text{ME} = 1.96 \times \frac{91}{\sqrt{112}}
\]

First, calculate the standard error (\( \frac{s}{\sqrt{n}} \)):

\[
\frac{91}{\sqrt{112}} \approx \frac{91}{10.583} \approx 8.60
\]

Now, calculate the margin of error:

\[
\text{ME} \approx 1.96 \times 8.60 \approx 16.85
\]

Rounding to the nearest whole number:

\[
\text{Margin of Error} \approx 17
\]

So, the margin of error is **17**.

Would you like more details or have any questions? Here are some related questions to consider:

1. How would the margin of error change with a larger sample size?
2. What if the confidence level was 90% instead of 95%? How would the margin of error change?
3. Can you explain why the z-score is used in this calculation?
4. How does the standard deviation affect the margin of error?
5. What assumptions are made when calculating the margin of error in this way?

**Tip:** The margin of error decreases as the sample size increases, which leads to more precise estimates of the population mean.

A study of a local high school tried to determine the mean number of text messages that each student sent per day. The study surveyed a random sample of 112 students in the high school and found a mean of 191 messages sent per day with a standard deviation of 91 messages. At the 95% confidence level, find the margin of error for the mean, rounding to the nearest whole number. (Do not write plus or minus±).

Calculate the margin of error for the mean number of text messages sent per day by students, based on a random sample of 112 students with a mean of 191 messages and a standard deviation of 91. Learn how to apply the margin of error formula at the 95% confidence level.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation