Suppose adults in a particular town drive a mean of 65 mph with a standard deviation of 15 mph. Using the normal curve table,
a.What percentage of adults drive below 35 mph?
b.What percentage of adults drive below 68 mph or above 85 mph?
c.What percentage of adults drive between 56 and 80 mph?
d.At what speed does someone need to drive to be included in the top 10%?
e.At what speed does someone need to drive to be included in the bottom 25%?
To answer these questions, we can use the Z-score formula:
Z = (X - μ) / σ
Where:
- X represents the value of interest
- μ represents the mean
- σ represents the standard deviation
Let's proceed with each question step by step:
a. What percentage of adults drive below 35 mph?
To find this, we need to calculate the Z-score using the formula:
Z = (35 - 65) / 15 = -2
Using the normal curve table, we can find that the area to the left of Z = -2 is 0.0228 or 2.28%.
Therefore, approximately 2.28% of adults drive below 35 mph.
b. What percentage of adults drive below 68 mph or above 85 mph?
To determine this, we need to find the individual probabilities of driving below 68 mph and above 85 mph and then sum them.
For below 68 mph:
Z = (68 - 65) / 15 = 0.2
Using the normal curve table, we find that the area to the left of Z = 0.2 is 0.5793 or 57.93%.
For above 85 mph:
Z = (85 - 65) / 15 = 1.33
Using the normal curve table, we find that the area to the left of Z = 1.33 is 0.9080 or 90.80%.
Therefore, the percentage of adults driving below 68 mph or above 85 mph is:
57.93% + (1 - 90.80%) = 57.93% + 9.20% = 67.13%.
Approximately 67.13% of adults drive below 68 mph or above 85 mph.
c. What percentage of adults drive between 56 and 80 mph?
To determine this, we need to calculate the individual probabilities of driving below 80 mph and below 56 mph, and then subtract the latter from the former.
For below 80 mph:
Z = (80 - 65) / 15 = 1
Using the normal curve table, we find that the area to the left of Z = 1 is 0.8413 or 84.13%.
For below 56 mph:
Z = (56 - 65) / 15 = -0.6
Using the normal curve table, we find that the area to the left of Z = -0.6 is 0.2743 or 27.43%.
Therefore, the percentage of adults driving between 56 and 80 mph is:
84.13% - 27.43% = 56.70%.
Approximately 56.70% of adults drive between 56 and 80 mph.
d. At what speed does someone need to drive to be included in the top 10%?
To find this, we need to determine the Z-score corresponding to the top 10% of the data.
Using the normal curve table, we find that the Z-score that corresponds to the top 10% is approximately 1.28.
We can solve for X in the Z-score formula:
1.28 = (X - 65) / 15
Rearranging the equation, we get:
X - 65 = 1.28 * 15
X - 65 = 19.2
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. At what speed does someone need to drive to be included in the bottom 25%?
To find this, we need to determine the Z-score that corresponds to the bottom 25% of the data.
Using the normal curve table, we find that the Z-score corresponding to the bottom 25% is approximately -0.67.
We can solve for X in the Z-score formula:
-0.67 = (X - 65) / 15
Rearranging the equation, we get:
X - 65 = -0.67 * 15
X - 65 = -10.05
X = 65 - 10.05
X ≈ 54.95
Therefore, someone needs to drive at approximately 54.95 mph to be included in the bottom 25%.
To answer these questions, we can use the normal distribution and the z-score. The z-score tells us how many standard deviations an individual value is away from the mean.
a. To find the percentage of adults driving below 35 mph, we need to find the z-score for 35 using the formula z = (x - mean) / standard deviation.
z = (35 - 65) / 15 = -2
Using the normal distribution table, we can find that the percentage of adults driving below 35 mph is approximately 0.0228 or 2.28%.
b. To find the percentage of adults driving below 68 mph or above 85 mph, we need to find the z-scores for these two speeds.
For 68 mph:
z = (68 - 65) / 15 = 0.2
For 85 mph:
z = (85 - 65) / 15 = 1.33
Using the normal distribution table, we can find the cumulative probabilities for these z-scores.
P(z < 0.2) = 0.5793
P(z > 1.33) = 0.9088
To find the percentage of adults driving below 68 mph or above 85 mph, we need to add these probabilities together and subtract it from 1.
P(68 mph or above 85 mph) = 1 - (P(z < 0.2) + P(z > 1.33))
= 1 - (0.5793 + 0.9088)
= 0.5119 or 51.19%
c. To find the percentage of adults driving between 56 and 80 mph, we need to find the z-scores for these two speeds.
For 56 mph:
z = (56 - 65) / 15 = -0.6
For 80 mph:
z = (80 - 65) / 15 = 1
Using the normal distribution table, we can find the cumulative probabilities for these z-scores.
P(z < -0.6) = 0.2743
P(z < 1) = 0.8413
To find the percentage of adults driving between 56 and 80 mph, we subtract the probabilities.
P(56 mph to 80 mph) = P(z < 1) - P(z < -0.6)
= 0.8413 - 0.2743
= 0.567 or 56.7%
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 top 10% or 1 - 0.10 = 0.90.
Using the normal distribution table, we can find the z-score that has a cumulative probability of 0.90. The closest value is approximately 1.28.
Using the formula z = (x - mean) / standard deviation, we can solve for x:
1.28 = (x - 65) / 15
Solving for x, we get:
x - 65 = 1.28 * 15
x - 65 = 19.2
x = 84.2
So, someone needs to drive at least 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 bottom 25% or 0.25.
Using the normal distribution table, we can find the z-score that has a cumulative probability of 0.25. The closest value is approximately -0.67.
Using the formula z = (x - mean) / standard deviation, we can solve for x:
-0.67 = (x - 65) / 15
Solving for x, we get:
x - 65 = -0.67 * 15
x - 65 = -10.05
x = 54.95
So, someone needs to drive at most 54.95 mph to be included in the bottom 25%.