A population has mean [tex]$\mu=10$[/tex] and standard deviation [tex]$\sigma=3$[/tex]. Round the answers to two decimal places as needed.
(a) Find the [tex]$z$[/tex]-score for a population value of 1.
The [tex]$z$[/tex]-score for a population value of 1 is ______.
Answer (1)
Calculate the z -score using the formula: z = σ x − μ .
Substitute the given values: z = 3 1 − 10 .
Calculate the z -score: z = − 3 .
The z -score for a population value of 1 is − 3 .
Explanation
Understand the problem and provided data We are given a population with a mean μ = 10 and a standard deviation σ = 3 . We want to find the z -score for a population value of x = 1 . The z -score tells us how many standard deviations away from the mean a particular data point is.
Recall the z-score formula The formula for calculating the z -score is: z = σ x − μ where:
x is the population value,
μ is the population mean,
σ is the population standard deviation.
Substitute the values and calculate Now, we substitute the given values into the formula: z = 3 1 − 10 z = 3 − 9 z = − 3
State the final answer The z -score for a population value of 1 is − 3 . This means that the value 1 is 3 standard deviations below the mean of the population.
Examples
The z-score is a measure of how many standard deviations an individual data point is from the mean of a distribution. For example, in a classroom, if the average test score is 75 with a standard deviation of 5, a student who scored 85 would have a z-score of 2, indicating their score is significantly above average. Z-scores are used in many statistical applications, such as hypothesis testing and constructing confidence intervals. Understanding z-scores helps in comparing data points from different distributions and assessing their relative positions.
