When fifteen vehicles were observed at random for their speeds (in M.P.H.) on a freeway, the following data were obtained.
61 53 57 55 62 58 60 54 62 60 59 66 55 54 63
Suppose that from past experience it may be assumed that vehicle speed is normally distributed with standard deviation of 3.96. Construct a 93 percent confidence interval for the true mean speed.
To construct a confidence interval, we need to know the sample mean, sample standard deviation, sample size, and the desired level of confidence.
1. Calculate the sample mean:
Add up all the speeds and divide by the sample size:
Sample mean = (61 + 53 + 57 + 55 + 62 + 58 + 60 + 54 + 62 + 60 + 59 + 66 + 55 + 54 + 63) / 15 = 59.2
2. Calculate the sample standard deviation:
Subtract the sample mean from each speed, square the differences, sum them, divide by the sample size minus 1, and take the square root:
Subtract the sample mean from each speed:
(61-59.2), (53-59.2), (57-59.2), (55-59.2), (62-59.2), (58-59.2), (60-59.2), (54-59.2), (62-59.2), (60-59.2), (59-59.2), (66-59.2), (55-59.2), (54-59.2), (63-59.2)
Square the differences:
(2.56), (36.00), (5.76), (18.36), (7.84), (1.44), (0.64), (25.00), (7.84), (1.44), (0.04), (41.16), (18.36), (25.00), (11.56)
Sum the squared differences:
221.08
Divide by sample size minus 1:
221.08 / (15-1) = 15.79
Take the square root:
Sample standard deviation = sqrt(15.79) = 3.97
3. Determine the desired level of confidence:
The problem states we want a 93 percent confidence interval.
4. Calculate the critical z-value:
Since the distribution is assumed to be normally distributed and we have a large enough sample size (n > 30), we can use the Z-distribution.
To find the critical z-value, we need to find the z-value corresponding to 93 percent confidence level.
Look up the critical z-value in a standard normal distribution table or use a calculator:
For a 93 percent confidence interval, the critical z-value is approximately 1.81.
5. Calculate the margin of error:
The margin of error is the product of the critical z-value and the standard deviation of the sample mean.
Margin of error = critical z-value * (sample standard deviation / sqrt(sample size))
Margin of error = 1.81 * (3.97 / sqrt(15))
Margin of error ≈ 1.12
6. Calculate the confidence interval:
The confidence interval is calculated by subtracting and adding the margin of error to the sample mean.
Confidence interval = sample mean ± margin of error
Confidence interval = 59.2 ± 1.12
7. Final result:
Confidence interval for the true mean speed = (59.2 - 1.12, 59.2 + 1.12)
Confidence interval for the true mean speed ≈ (58.08, 60.32)
Therefore, we can be 93 percent confident that the true mean speed lies between 58.08 mph and 60.32 mph.