Assume a data set is normally distributed with mean 11 and standard deviation 3. If the data set contains 30 data values, approximately how many of the data values will fall within the range 8 to 14?
To determine the number of data values that fall within the range of 8 to 14 in a normally distributed data set with mean 11 and standard deviation 3, we can use the empirical rule (also known as the 68-95-99.7 rule) for a normal distribution.
According to the empirical rule, approximately 68% of the data values will fall within one standard deviation of the mean. In this case, the range between 8 and 14 falls within one standard deviation of the mean.
Therefore, approximately 68% of the data values in the set will fall within the range of 8 to 14.
To find the approximate number of data values within this range, we can use the formula:
Number of data values = (Percentage / 100) * Total number of data values
Number of data values = (68 / 100) * 30
Number of data values ≈ 0.68 * 30
Number of data values ≈ 20.4
Thus, approximately 20 data values will fall within the range of 8 to 14 in the data set.
To find the approximate number of data values that fall within a specific range, we need to use the properties of the normal distribution.
In this case, we know that the data set is normally distributed with a mean of 11 and a standard deviation of 3.
First, we'll find the z-scores for the lower and upper boundaries of the range. The z-score measures the number of standard deviations a particular value is from the mean.
The formula to calculate the z-score is:
z = (x - μ) / σ
Where:
- x is the value we want to find the z-score for,
- μ is the mean of the distribution,
- σ is the standard deviation of the distribution.
For the lower boundary, x = 8:
z_lower = (8 - 11) / 3 = -1
For the upper boundary, x = 14:
z_upper = (14 - 11) / 3 = 1
Next, we use a standard normal distribution table or a calculator to find the probability of the values falling within the z-scores.
The table or calculator gives us the cumulative probability, which is the probability of a value being less than or equal to a certain z-score.
Using the table, we find that the probability of a value being less than or equal to -1 is approximately 0.1587, and the probability of a value being less than or equal to 1 is approximately 0.8413.
To find the probability of values falling within the range, we subtract the lower cumulative probability from the upper cumulative probability:
P = 0.8413 - 0.1587
≈ 0.6826
So approximately 68.26% of the data values will fall within the range 8 to 14.
To find the approximate number of data values within the range, we multiply the probability by the total number of data values:
Number of data values ≈ 0.6826 * 30
≈ 20.478
Therefore, approximately 20 data values will fall within the range 8 to 14.