In a set of 400 values, the mean is 121 and the standard deviation is 14. Estimate how many values fall in between each interval.
a) 107 - 135 =
b) 93 - 149 =
c) 79 - 163 =
a)24
b)56
c)84
The answer is C because it is 399 values found and it's close to 400 values
To estimate the number of values that fall within each interval, you need to use the properties of the normal distribution and Z-scores.
a) To estimate the number of values between 107 and 135, you need to find the Z-score for each boundary value and then use a Z-table (standard normal distribution table) to determine the corresponding probabilities.
The Z-score formula is:
Z = (X - μ) / σ
Where:
- Z is the Z-score
- X is the value you want to find the probability for
- μ is the mean of the distribution
- σ is the standard deviation of the distribution
For the lower boundary:
Z1 = (107 - 121) / 14 = -14 / 14 = -1
For the upper boundary:
Z2 = (135 - 121) / 14 = 14 / 14 = 1
Now, you need to find the probabilities corresponding to these Z-scores from the Z-table. For Z = -1, the probability is 0.1587, and for Z = 1, the probability is 0.8413.
To estimate the number of values between 107 and 135, you subtract the lower probability from the upper probability and then multiply the result by the total number of values (400):
Number of values = (0.8413 - 0.1587) * 400 = 0.6826 * 400 ≈ 273
Therefore, the estimate for the number of values between 107 and 135 is approximately 273.
b) The process is the same for the interval 93 - 149. You need to find the Z-scores for 93 and 149, and then calculate the probabilities and multiply them by the total number of values (400) to get the estimate.
For the lower boundary:
Z1 = (93 - 121) / 14 = -28 / 14 = -2
For the upper boundary:
Z2 = (149 - 121) / 14 = 28 / 14 = 2
Using the Z-table, you find that the probabilities for Z = -2 and Z = 2 are approximately 0.0228 and 0.9772, respectively.
Number of values = (0.9772 - 0.0228) * 400 = 0.9544 * 400 ≈ 382
Therefore, the estimate for the number of values between 93 and 149 is approximately 382.
c) The process is the same for the interval 79 - 163. Find the Z-scores for 79 and 163, and then calculate the probabilities and multiply them by the total number of values (400) to get the estimate.
For the lower boundary:
Z1 = (79 - 121) / 14 = -42 / 14 = -3
For the upper boundary:
Z2 = (163 - 121) / 14 = 42 / 14 = 3
Using the Z-table, you find that the probabilities for Z = -3 and Z = 3 are approximately 0.0013 and 0.9987, respectively.
Number of values = (0.9987 - 0.0013) * 400 = 0.9974 * 400 ≈ 399
Therefore, the estimate for the number of values between 79 and 163 is approximately 399.