The authors of a certain paper describe a study to evaluate the effect of mobile phone use by taxi drivers in Greece. Fifty taxi drivers drove in a driving simulator where they were following a lead car. The drivers were asked to carry on a conversation on a mobile phone while driving, and the following distance (the distance between the taxi and the lead car) was recorded.
The sample mean following distance was 3.40 meters and the sample standard deviation was 1.13 meters.
(a) Construct a 95% confidence interval (in meters) for 𝜇, the population mean following distance while talking on a mobile phone for the population of taxi drivers.
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To construct a confidence interval for the population mean following distance while talking on a mobile phone, we can use the formula:
Confidence Interval = sample mean +/- (critical value * standard error)
First, we need to calculate the standard error using the formula:
Standard Error = sample standard deviation / √(sample size)
Given:
Sample mean = 3.40 meters
Sample standard deviation = 1.13 meters
Sample size = 50
Standard Error = 1.13 / √(50) = 1.13 / 7.071 = 0.16
Next, we need to find the critical value for a 95% confidence interval. Since we have a large sample size (n > 30), we can use the z-distribution. The critical value for a 95% confidence interval is 1.96.
Confidence Interval = 3.40 +/- (1.96 * 0.16)
Lower limit = 3.40 - (1.96 * 0.16) = 3.40 - 0.3136 = 3.0864 meters
Upper limit = 3.40 + (1.96 * 0.16) = 3.40 + 0.3136 = 3.7136 meters
Therefore, the 95% confidence interval for the population mean following distance while talking on a mobile phone for the population of taxi drivers is [3.0864 meters, 3.7136 meters].
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