Math Problem Statement

A research group wishes to estimate the mean amount of time (in hours) that members of a fitness center spend exercising each week. They want to estimate the mean with a margin of error 0.5 hours with 96% level of confidence. Previous data suggests that the standard deviation of the population is 2.12 hours.

Solution

To calculate the required sample size (nn) needed to estimate the mean time spent exercising each week, we use the formula for the margin of error of a confidence interval:

E=zσnE = z \cdot \frac{\sigma}{\sqrt{n}}

Where:

  • EE is the margin of error (0.50.5 hours in this case),
  • zz is the critical value corresponding to the desired confidence level (96%96\%),
  • σ\sigma is the population standard deviation (2.122.12 hours),
  • nn is the required sample size.

Step 1: Determine the critical value (zz)

For a 96%96\% confidence level, the area in each tail is (10.96)/2=0.02(1 - 0.96)/2 = 0.02. Using a standard normal table or z-score calculator:

z2.05z \approx 2.05

Step 2: Solve for nn

Rearrange the formula to solve for nn:

n=(zσE)2n = \left(\frac{z \cdot \sigma}{E}\right)^2

Substitute the values: z=2.05,σ=2.12,E=0.5z = 2.05, \quad \sigma = 2.12, \quad E = 0.5

n=(2.052.120.5)2n = \left(\frac{2.05 \cdot 2.12}{0.5}\right)^2

n=(4.3460.5)2n = \left(\frac{4.346}{0.5}\right)^2

n=(8.692)275.56n = (8.692)^2 \approx 75.56

Step 3: Round up

Since the sample size must be a whole number, round up to the nearest integer:

n=76n = 76

Final Answer:

The research group needs a sample size of 76 to estimate the mean with a margin of error of 0.5 hours and a 96%96\% confidence level.


Do you want further explanation or have any questions?

Here are 5 related questions to consider:

  1. How does increasing the confidence level affect the sample size?
  2. What happens if the margin of error is reduced to 0.25 hours?
  3. Why do we round up the sample size instead of rounding to the nearest integer?
  4. How would this calculation change if the population standard deviation was unknown?
  5. Can this method be used if the population distribution is not normal?

Tip: Always use the critical z-value corresponding to your specific confidence level to ensure accuracy in sample size calculations.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Statistics
Confidence Intervals
Sample Size Estimation

Formulas

E = z * (σ / √n)
n = (z * σ / E)^2

Theorems

Properties of Normal Distribution

Suitable Grade Level

Undergraduate or Advanced High School (Grades 11-12)