A set of 50 data values has a mean of 27 and a variance of 16.
I. Find the standard score (z) for a data value = 19.
II. Find the probability of a data value < 19.
III. Find the probability of a data value > 19.
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steps to finding variance of sample data set.11 10 16 18 5 17 8 6 7
To find the answers to these questions, we'll use the concept of standard deviation and the standard normal distribution.
I. To find the standard score (z-score) for a data value of 19, we'll use the formula:
z = (x - μ) / σ
Where:
- x represents the data value
- μ represents the mean
- σ represents the standard deviation
Given:
- x = 19
- μ = 27
- σ = √16 = 4
Applying these values to the formula, we get:
z = (19 - 27) / 4
z = -8 / 4
z = -2
Therefore, the standard score (z) for a data value of 19 is -2.
II. To find the probability of a data value less than 19, we'll convert the value to a z-score and then use a standard normal distribution table or calculator to find the corresponding probability.
Using the z-score from the previous part (z = -2), we look up the corresponding probability in the standard normal distribution table.
From the standard normal distribution table, we find that the probability corresponding to z = -2 is approximately 0.0228.
Therefore, the probability of a data value less than 19 is approximately 0.0228.
III. To find the probability of a data value greater than 19, we can use the complement rule. The complement of "greater than 19" is "less than or equal to 19". We can calculate the probability of a value less than or equal to 19 and then subtract it from 1.
Using the z-score from the previous part (z = -2), we can look up the probability of a value less than or equal to -2 in the standard normal distribution table. From the table, we find that the probability is approximately 0.0228.
Now, subtracting this value from 1:
P(X > 19) = 1 - P(X ≤ 19)
= 1 - 0.0228
= 0.9772
Therefore, the probability of a data value greater than 19 is approximately 0.9772.