The Florida Department of Transportation (FDOT) would like to estimate the average speed of vehicles on I95. The mean speed, in miles per hour, of a random sample of vehicles was found to be 89 miles per hour. The standard deviation of the speeds of all vehicles on 1.95 is known to be 14 miles per hour.
Find three different confidence intervals - one with sample size 255 , one with sample size 387 , and one with sample size 549 . Assume that the mean of each sample is 89 miles per hour. Notice how the sample size affects the margin of error and the width of the interval.
Report confidence interval solutions using interval notation. Round solutions to two decimal places, if necessary.
- When $n=255$, the margin of error for a $97 \%$ confidence interval is given by
When $n=255$, a $97 \%$ confidence interval is given by
- When $n=387$, the margin of error for a $97 \%$ confidence interval is given by
When $n=387$, a $97 \%$ confidence interval is given by
- When $n=549$, the margin of error for a $97 \%$ confidence interval is given by
When $n=549$, a $97 \%$ confidence interval is given by
If the sample size is increased, leaving all other characteristics constant, the margin of error of the confidence interval will Select an answer $v$.
If the sample size is increased, leaving all other characteristics constant, the width of the confidence interval will Select an answer - .
Answer
Final Answer: \n- When \(n=255\), a \(97 \%\) confidence interval is given by \(\boxed{(87.10, 90.90)}\)\n- When \(n=387\), a \(97 \%\) confidence interval is given by \(\boxed{(87.46, 90.54)}\)\n- When \(n=549\), a \(97 \%\) confidence interval is given by \(\boxed{(87.70, 90.30)}\)\nIf the sample size is increased, leaving all other characteristics constant, the margin of error of the confidence interval will decrease.\nIf the sample size is increased, leaving all other characteristics constant, the width of the confidence interval will decrease.
Step 1 ：The problem is asking for three different confidence intervals for the average speed of vehicles on I95, given a mean speed of 89 miles per hour and a standard deviation of 14 miles per hour. The sample sizes are 255, 387, and 549. The confidence level is 97%.
Step 2 ：The formula for a confidence interval is given by: mean ± Z * (std_dev / sqrt(n)) where: mean is the sample mean, Z is the Z-score, which corresponds to the desired confidence level, std_dev is the standard deviation, and n is the sample size.
Step 3 ：The Z-score for a 97% confidence level is approximately 2.33.
Step 4 ：For n=255, the 97% confidence interval is approximately (87.10, 90.90).
Step 5 ：For n=387, the 97% confidence interval is approximately (87.46, 90.54).
Step 6 ：For n=549, the 97% confidence interval is approximately (87.70, 90.30).
Step 7 ：As the sample size increases, the margin of error decreases, which means the confidence interval becomes narrower. This is because a larger sample size provides more information about the population, which reduces the uncertainty (and thus the margin of error) in our estimate of the population mean.
Step 8 ：Final Answer: \n- When \(n=255\), a \(97 \%\) confidence interval is given by \(\boxed{(87.10, 90.90)}\)\n- When \(n=387\), a \(97 \%\) confidence interval is given by \(\boxed{(87.46, 90.54)}\)\n- When \(n=549\), a \(97 \%\) confidence interval is given by \(\boxed{(87.70, 90.30)}\)\nIf the sample size is increased, leaving all other characteristics constant, the margin of error of the confidence interval will decrease.\nIf the sample size is increased, leaving all other characteristics constant, the width of the confidence interval will decrease.
