Answer:
Explanation:
To solve these questions, we can use the Z-score formula and the standard normal distribution table (also known as the normal curve table). The Z-score formula is given by:
Z = (X - μ) / σ
Where:
Z is the Z-score,
X is the observed value,
μ is the mean, and
σ is the standard deviation.
(a) To find the percentage of adults driving below 35 mph, we need to calculate the Z-score for 35 mph and find the corresponding area under the normal curve table.
Z = (35 - 65) / 15 = -2
Looking up the Z-score of -2 in the normal curve table, we find that the area to the left of -2 is approximately 0.0228 or 2.28%. Therefore, approximately 2.28% of adults drive below 35 mph.
(b) To find the percentage of adults driving below 68 mph or above 85 mph, we need to calculate the Z-scores for both values and add the corresponding areas under the normal curve table.
For 68 mph:
Z1 = (68 - 65) / 15 = 0.2
For 85 mph:
Z2 = (85 - 65) / 15 = 1.33
Now, we look up the Z-scores 0.2 and 1.33 in the normal curve table. The area to the left of 0.2 is approximately 0.5793 or 57.93%, and the area to the left of 1.33 is approximately 0.9088 or 90.88%.
To find the area below 68 mph or above 85 mph, we subtract the area to the left of 68 mph from 100% (1.0) and add the area to the right of 85 mph.
Area = (1.0 - 0.5793) + (1.0 - 0.9088) = 0.4207 + 0.0912 = 0.5119
Therefore, approximately 51.19% of adults drive below 68 mph or above 85 mph.
(c) To find the percentage of adults driving between 56 and 80 mph, we need to calculate the Z-scores for both values and find the difference between their areas under the normal curve table.
For 56 mph:
Z1 = (56 - 65) / 15 = -0.6
For 80 mph:
Z2 = (80 - 65) / 15 = 1
Now, we look up the Z-scores -0.6 and 1 in the normal curve table. The area to the left of -0.6 is approximately 0.2743 or 27.43%, and the area to the left of 1 is approximately 0.8413 or 84.13%.
To find the area between 56 mph and 80 mph, we subtract the area to the left of 56 mph from the area to the left of 80 mph.
Area = 0.8413 - 0.2743 = 0.567
Therefore, approximately 56.7% of adults drive between 56 and 80 mph.
(d) To find the speed at which someone needs to drive to be included in the top 10%, we need to find the Z-score that corresponds to the area of 90% (100% - 10%) under the normal curve table.
Looking up the Z-score for an area of 90% in the normal curve table, we find that the Z-score is approximately 1.28.
Now, we can use the Z-score formula to find the corresponding speed:
Z = (X - μ) / σ
1.28 = (X - 65) / 15
Solving for X, we have:
X - 65 = 1.28 * 15
X - 65 = 19.2
X = 84.2
Therefore, someone needs to drive at approximately 84.2 mph to be included in the top 10%.
(e) To find the speed at which someone needs to drive to be included in the bottom 25%, we need to find the Z-score that corresponds to the area of 25% under the normal curve table.
Looking up the Z-score for an area of 25% in the normal curve table, we find that the Z-score is approximately -0.67.
Now, we can use the Z-score formula to find the corresponding speed:
Z = (X - μ) / σ
-0.67 = (X - 65) / 15
Solving for X, we have:
X - 65 = -0.67 * 15
X - 65 = -10.05
X = 54.95
Therefore, someone needs to drive at approximately 54.95 mph to be included in the bottom Below is a summary of the answers:
(a) Approximately 2.28% of adults drive below 35 mph.
(b) Approximately 51.19% of adults drive below 68 mph or above 85 mph.
(c) Approximately 56.7% of adults drive between 56 and 80 mph.
(d) Someone needs to drive at approximately 84.2 mph to be included in the top 10%.
(e) Someone needs to drive at approximately 54.95 mph to be included in the bottom 25%.