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 86 students in the high school and found a mean of 160 messages sent per day with a standard deviation of 53 messages. Determine a 95% confidence interval for the mean, rounding all values to the nearest whole number.
To determine the 95% confidence interval for the mean number of text messages sent per day by high school students, we can use the formula:
Confidence Interval = (Sample Mean) ± (Z * Standard Deviation / √n)
Where:
- Sample Mean: 160 messages
- Standard Deviation: 53 messages
- n: Sample Size (86 students)
- Z: Critical value for a 95% confidence level (we'll use the standard normal distribution)
Let's calculate the confidence interval step by step:
1. Look up the critical value for a 95% confidence level.
The critical value can be found using a standard normal distribution table or a statistical calculator. For a 95% confidence level, the critical value is approximately 1.96.
2. Calculate the standard error (Standard Deviation / √n).
Standard error = 53 / √86 ≈ 5.71
3. Calculate the margin of error (Z * Standard Error).
Margin of error = 1.96 * 5.71 ≈ 11.19 (round to two decimal places)
4. Calculate the lower and upper bounds of the confidence interval.
Lower bound = Sample Mean - Margin of error = 160 - 11.19 ≈ 148.81 (round to the nearest whole number)
Upper bound = Sample Mean + Margin of error = 160 + 11.19 ≈ 171.19 (round to the nearest whole number)
Therefore, the 95% confidence interval for the mean number of text messages sent per day by high school students is approximately 149 to 171 messages.