Researchers want to estimate the mean time people spend grocery shopping per week. If they want the margin of error to be within 1.5 hour, how many people should they survey? Assume that they want to calculate a 95% confidence interval and the population standard deviation is about 5.8 hours.
Question
Researchers want to estimate the mean time people spend grocery shopping per week. If they want the margin of error to be within 1.5 hour, how many people should they survey? Assume that they want to calculate a 95% confidence interval and the population standard deviation is about 5.8 hours.
Solution
To calculate the sample size needed, we can use the formula for the sample size in estimating a population mean:
n = (Z^2 * σ^2) / E^2
where:
- n is the sample size
- Z is the Z-score, which corresponds to the desired confidence level (1.96 for a 95% confidence interval)
- σ is the population standard deviation (5.8 hours in this case)
- E is the desired margin of error (1.5 hours in this case)
Substituting the given values into the formula, we get:
n = (1.96^2 * 5.8^2) / 1.5^2
Solving this equation will give us the required sample size.
Please note that the result may not be a whole number, but since we can't survey a fraction of a person, we'll need to round up to the nearest whole number.
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