Assume a data set is normally distributed with mean 182 and standard deviation 30. If the data set contains 600 data values, approximately how many of the data values will fall within the range 122 to 242?
122
z = (122-182)/30 = -2
242
z = (242 - 182)/30 = +2
then use normal table
or use:
for example
http://davidmlane.com/hyperstat/z_table.html
result .9545 in that range so
.9545*600 = 573
To calculate the number of data values that will fall within a given range in a normal distribution, we need to use the properties of the standard normal distribution and the Z-score.
The Z-score represents how many standard deviations a given value is from the mean. It can be calculated using the formula:
Z = (X - μ) / σ
Where:
- X is the given value,
- μ is the mean of the distribution, and
- σ is the standard deviation of the distribution.
In this case, we want to find the number of data values that fall within the range 122 to 242. To convert these values into Z-scores, we use the formula for each value:
Z1 = (122 - μ) / σ
Z2 = (242 - μ) / σ
Substituting the given values, we have:
Z1 = (122 - 182) / 30
= -2
Z2 = (242 - 182) / 30
= 2
Now, we need to determine the probability that a data value falls within this range.
Using a standard normal distribution table or a calculator, we can find the cumulative probability for Z1 and Z2. The cumulative probability for Z1 represents the probability that a data value is less than 122, and the cumulative probability for Z2 represents the probability that a data value is less than or equal to 242.
Let's denote P1 as the cumulative probability for Z1 and P2 as the cumulative probability for Z2.
P1 ≈ 0.0228 (from the standard normal distribution table) ---> Probability that a data value is less than 122.
P2 ≈ 0.9772 (from the standard normal distribution table) ---> Probability that a data value is less than or equal to 242.
To find the probability that a data value falls within the range 122 to 242, we subtract P1 from P2:
P2 - P1 ≈ 0.9772 - 0.0228 ≈ 0.9544
The resulting probability, approximately 0.9544, represents the proportion of data values that fall within the range 122 to 242.
To find the actual number of data values within this range, we multiply this probability by the total number of data values (600):
Approximate number of data values = 0.9544 * 600 ≈ 572
Therefore, approximately 572 of the 600 data values will fall within the range 122 to 242.