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.
To answer these questions, we need to understand the concept of the standard score (z-score) and the standard normal distribution.
I. To find the standard score for a data value of 19, we use the formula: z = (x - mean) / standard deviation.
Since we are not provided with the standard deviation directly, we can find it using the variance. The standard deviation (σ) is the square root of the variance (σ^2).
Given that the variance is 16, we can calculate the standard deviation (σ) as follows: σ = √16 = 4.
Now we can calculate the z-score using the formula:
z = (19 - 27) / 4 = -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 need to convert it into a z-score and then use a standard normal distribution table or calculator.
Since we already calculated the z-score in the previous step as -2, we can now use a standard normal distribution table (also known as a z-table) to find the corresponding probability.
In the standard normal distribution table, the z-score of -2 corresponds to a probability of 0.0228. This means that the probability of a data value being less than 19 is approximately 0.0228, or 2.28%.
III. To find the probability of a data value greater than 19, we can subtract the probability of the data value being less than 19 from 1 (because the total probability is 1).
Using the z-score we calculated earlier (-2) and the corresponding probability from the standard normal distribution table (0.0228), we can calculate the probability of a data value greater than 19 as follows:
P(data value > 19) = 1 - P(data value < 19) = 1 - 0.0228 = 0.9772
Therefore, the probability of a data value being greater than 19 is approximately 0.9772, or 97.72%.