Statistic Question (Homework Help)

Question

Subject	1	2	3	4	5	6	7	8	9
​Normal, Xi	4.49	4.34	4.58	4.65	4.31	4.83	4.59	5.00	4.79
​Impaired, Yi	5.77	5.67	5.45	5.29	5.90	5.49	5.23	5.61	5.63

To illustrate the effects of driving under the influence​ (DUI) of​ alcohol, a police officer brought a DUI simulator to a local high school. Student reaction time in an emergency was measured with unimpaired vision and also while wearing a pair of special goggles to simulate the effects of alcohol on vision. For a random sample of nine​ teenagers, the time​ (in seconds) required to bring the vehicle to a stop from a speed of 60 miles per hour was recorded. Complete parts​ (a) and​ (b).

​Note: A normal probability plot and boxplot of the data indicate that the differences are approximately normally distributed with no outliers.

Question:

Use a​ 95% confidence interval to test if there is a difference in braking time with impaired vision and normal vision where the differences are computed as​ "impaired minus​ normal."

The lower bound is ?

The upper bound is ?

Round to the nearest thousandth as​ needed.)

Answer 1

To test if there is a difference in braking time with impaired vision and normal vision, you can perform a paired t-test. The differences between impaired and normal vision braking times are computed as "impaired minus normal."

First, compute the differences in braking times:

1. Impaired Vision (Yi): 5.77, 5.67, 5.45, 5.29, 5.90, 5.49, 5.23, 5.61, 5.63

2. Normal Vision (Xi): 4.49, 4.34, 4.58, 4.65, 4.31, 4.83, 4.59, 5.00, 4.79

Differences = Yi - Xi: (5.77 - 4.49), (5.67 - 4.34), (5.45 - 4.58), (5.29 - 4.65), (5.90 - 4.31), (5.49 - 4.83), (5.23 - 4.59), (5.61 - 5.00), (5.63 - 4.79)

Differences = 1.28, 1.33, 0.87, 0.64, 1.59, 0.66, 0.64, 0.61, 0.84

Now, calculate the sample mean (X̄) and the sample standard deviation (s) of these differences.

X̄ = (1.28 + 1.33 + 0.87 + 0.64 + 1.59 + 0.66 + 0.64 + 0.61 + 0.84) / 9

X̄ = 1.046 seconds (rounded to three decimal places)

s = √[Σ(xi - X̄)² / (n - 1)]

s = √[(Σ(1.28 - 1.046)² + (1.33 - 1.046)² + ... + (0.84 - 1.046)²) / (9 - 1)]

s ≈ 0.326 seconds (rounded to three decimal places)

Now, you can calculate the standard error of the mean (SE):

SE = s / √n

SE = 0.326 / √9

SE ≈ 0.108 seconds (rounded to three decimal places)

To construct a 95% confidence interval for the mean difference in braking times (impaired minus normal), use the t-distribution with 8 degrees of freedom (n - 1). The critical t-value for a 95% confidence interval (two-tailed) is approximately 2.306.

Confidence Interval (CI) = X̄ ± (t_critical * SE)

CI = 1.046 ± (2.306 * 0.108)

Lower bound: 1.046 - (2.306 * 0.108)

Lower bound ≈ 0.791 seconds

Upper bound: 1.046 + (2.306 * 0.108)

Upper bound ≈ 1.301 seconds

So, the lower bound of the 95% confidence interval is approximately 0.791 seconds, and the upper bound is approximately 1.301 seconds.