A set of 50 data values has a mean of 48 and a variance of 9.
I. Find the standard score (z) for a data value = 43.
II. Find the probability of a data value < 43.
Z = (score-mean)/SD
SD = √variance
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score.
Indicate your subject in the "School Subject" box, so those with expertise in the area will respond to the question. You lucked out this time.
To solve this problem, we'll need to use the concept of standard scores (also known as z-scores) and the standard normal distribution.
I. Finding the standard score (z) for a data value = 43:
The standard score (z-score) tells us how many standard deviations a particular data value is away from the mean. It is calculated using the formula:
z = (x - μ) / σ
where:
- x is the data value
- μ is the mean
- σ is the standard deviation
Here, we are given the mean (μ = 48) and the variance (σ^2 = 9). To find the standard deviation (σ), we take the square root of the variance:
σ = √(9) = 3
Now, we can calculate the z-score for the data value of 43:
z = (43 - 48) / 3 = -1.67
Therefore, the standard score (z) for a data value of 43 is -1.67.
II. Finding the probability of a data value < 43:
To find the probability of a data value less than 43, we need to convert it into the standard normal distribution (also called the z-distribution) and then find the corresponding probability using a z-table or a statistical calculator.
The standard normal distribution has a mean of 0 and a standard deviation of 1. We can convert the data value of 43 into the standard normal distribution using the z-score formula:
z = (x - μ) / σ
In this case, μ = 48 and σ = 3. Therefore:
z = (43 - 48) / 3 = -1.67
To find the probability of a data value less than 43, we need to find the area under the standard normal curve to the left of the z-score -1.67. This probability can be looked up on a z-table or computed using a statistical calculator.
Consulting a z-table or using a calculator, we find that the area to the left of -1.67 is approximately 0.0475.
Therefore, the probability of a data value less than 43 is approximately 0.0475 (or 4.75%).