Question

(a) For a normal distribution, find the z-score that cuts off the bottom 55.76% of all the z-scores. Round to 2 decimal places after the decimal point. answer: (b) For a normal distribution, find the z-score that cuts off the top 77.52% of area. Round to 2 decimal places after the decimal point. answer:

Answer 1

a) Z = 0.15

b) Z = -0.76

Explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

(a) For a normal distribution, find the z-score that cuts off the bottom 55.76% of all the z-scores.

Bottom 55.76% of the scores are the z-scores with a pvalue of 0.5576 and less. So the z-score that cuts off these scores is Z when X has a pvalue of 0.5576, which is Z = 0.145, rounded to two decimal places, Z = 0.15.

(b) For a normal distribution, find the z-score that cuts off the top 77.52% of area.

The top 77.52% of the scores are the z-scores with a pvalue of 1-0.7752 = 0.2248 and higher. So Z = -0.764 and higher. So the value that cuts off the top 77.52% is Z = -0.76.