A motel proprietor asks new guests who arrive by car where they started their journey from that day. After three months, he has collected data on 50 carloads of guests. He goes through his records and calculates the distance (in km) travelled by carload of guests. The mean is 195km with a standard deviation of 42.3km.A 95% confidence interval for the average distance travelled is closest to
Question
A motel proprietor asks new guests who arrive by car where they started their journey from that day. After three months, he has collected data on 50 carloads of guests. He goes through his records and calculates the distance (in km) travelled by carload of guests. The mean is 195km with a standard deviation of 42.3km.A 95% confidence interval for the average distance travelled is closest to
Solution
A confidence interval is a range of values, derived from a data set, that is likely to contain the value of an unknown population parameter. In this case, we are looking for a 95% confidence interval for the average distance travelled.
The formula for a confidence interval is:
CI = X ± Z*(σ/√n)
where: X is the sample mean, Z is the z-score (which corresponds to the desired level of confidence, 1.96 for 95% confidence), σ is the standard deviation, and n is the sample size.
Substituting the given values into the formula:
CI = 195 ± 1.96*(42.3/√50)
Calculating the expression inside the parentheses gives us:
CI = 195 ± 1.96*5.98
Calculating the multiplication gives us:
CI = 195 ± 11.72
So, the 95% confidence interval for the average distance travelled is from 183.28 km to 206.72 km. This means that we are 95% confident that the true population mean lies within this interval.
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