The middle 69% of a normally distributed population lies between what two standard scores? (Give your answers correct to two decimal places.)
0.69 and 0.31
Between Z = -1 and Z = +1 are 68.26% of the scores.
The middl 55 of normally distributed population lies between what two standard scores?
To find the two standard scores that represent the middle 69% of a normally distributed population, we can use the concept of z-scores.
First, let's understand what a z-score is. A z-score measures the number of standard deviations a particular value is away from the mean of a distribution. It is calculated using the formula:
z = (x - μ) / σ
where:
- z is the z-score
- x is the value of interest
- μ is the mean of the distribution
- σ is the standard deviation of the distribution
For a standard normal distribution (mean = 0, standard deviation = 1), we can convert z-scores to percentiles using a z-table or a statistical calculator.
In this case, we are given the proportion (0.69) that represents the middle 69% of the distribution.
To find the first standard score, we need to find the z-score corresponding to the lower end of the middle 69%. We can use the cumulative distribution function (CDF) to do this.
Using a z-table or a statistical calculator with the entered proportion of 0.69, we find that the z-score corresponding to the lower end is approximately 0.31.
Next, to find the second standard score, we subtract the proportion from 1 to get the upper end of the middle 69%. So, 1 - 0.69 = 0.31.
Therefore, the two standard scores that define the middle 69% of the population lie between 0.31 and -0.31 (when rounded to two decimal places).