Assume a data set is normally distributed with mean 129 and standard deviation 35. If the data set
Contains 350 data values, approximately how many of the data values will fall within the range 59 to
199?
A)88
B)123
C)334
D)239

Question

Assume a data set is normally distributed with mean 129 and standard deviation 35. If the data set
Contains 350 data values, approximately how many of the data values will fall within the range 59 to
199? 
A)88 
B)123 
C)334 
D)239

Quizplus · Accepted Answer

To solve the problem, first we standardize the range 59 to 199 using the formula:
z = (x - μ) / σ

For the lower bound, we have:
z = (59 - 129) / 35 = -2

For the upper bound, we have:
z = (199 - 129) / 35 = 2

We can then use a standard normal distribution table or calculator to find the area under the curve between -2 and 2, which is approximately 0.9545.

Finally, we multiply this area by the total number of data values to get:
0.9545 x 350 = 334.18

Since we can't have a fraction of a data value, we round to the nearest whole number and get 334. Therefore, the answer is C. We can use the empirical rule to estimate the proportion of data values that fall within the range of 59 to 199. The empirical rule states that approximately 68% of data values fall within one standard deviation of the mean, 95% within two standard deviations of the mean, and 99.7% within three standard deviations of the mean for normally distributed data.

To apply this rule, we first need to find the z-scores for the values 59 and 199:

z1 = (59 - 129) / 35 = -2
z2 = (199 - 129) / 35 = 2

Next, we find the area under the standard normal distribution curve between these two z-scores:

area = P(-2 ≤ Z ≤ 2) ≈ 0.9545

Finally, we can estimate the proportion of data values in the range of 59 to 199 by multiplying this area by the total number of data values:

proportion = 0.9545 * 350 ≈ 334

Therefore, approximately 334 data values will fall within the range of 59 to 199. The best choice is C.

Assume a Data Set Is Normally Distributed with Mean 129