A set of 750 values has a normal distribution with a mean of 12.5 and a standard deviation of 0.36.
A. What percent of the data is between 12.25 and 12.75?
ANSWER: 51.6%
B. Find the interval about the mean within wich 45% of the data lie.
ANSWER: 12.28-12.72
C. Find the probability of that a value selected at random from this data is between 11.67 and 13.33.
ANSWER: 97.9%
Are these correct?
Yes, they are correct.
I find minor differences in the last digits, probably due to round-off or interpolation errors.
To verify the answers, we can use the properties of the normal distribution and perform calculations using the given mean and standard deviation.
A. To find the percentage of data between 12.25 and 12.75, we need to calculate the z-scores for these values and then find the area under the normal curve between these z-scores. The z-score formula is given by z = (x - mean) / standard deviation.
For 12.25:
z1 = (12.25 - 12.5) / 0.36 = -0.69
For 12.75:
z2 = (12.75 - 12.5) / 0.36 = 0.69
Using a standard normal distribution table or a calculator, we can look up the area between these z-scores, which is approximately 0.495. Therefore, the percentage of data between 12.25 and 12.75 is 0.495 * 100 = 49.5%.
So the answer is incorrect. It should be 49.5%.
B. To find the interval about the mean within which 45% of the data lie, we need to determine the z-scores such that the area between them accounts for 45% of the data.
Since it is a continuous distribution, we can use the standard normal distribution table or a calculator to determine the z-scores that correspond to the cumulative probabilities of 0.275 (0.5 - 0.45/2) and 0.725 (0.5 + 0.45/2). These z-scores represent the boundaries of the interval.
Using the table or calculator, we find z1 ≈ -1.76 and z2 ≈ 1.76.
To find the corresponding values in the original distribution, we use the formula x = mean + z * standard deviation:
For the lower boundary, x1 = 12.5 + (-1.76) * 0.36 ≈ 11.96
For the upper boundary, x2 = 12.5 + 1.76 * 0.36 ≈ 13.04
Thus, the interval is approximately 11.96-13.04.
So the answer is incorrect. It should be 11.96-13.04.
C. To find the probability that a value selected at random from this data is between 11.67 and 13.33, we need to calculate the corresponding z-scores and find the area between them.
For 11.67:
z1 = (11.67 - 12.5) / 0.36 ≈ -2.31
For 13.33:
z2 = (13.33 - 12.5) / 0.36 ≈ 2.31
Using the standard normal distribution table or a calculator, we can find the area between these z-scores, which is approximately 0.979. Thus, the probability is 0.979 * 100 = 97.9%.
So the answer is correct. The probability is approximately 97.9%.
In summary, the correct answers are:
A. 49.5%
B. 11.96-13.04
C. 97.9%