Answer:
a) The boundaries are
b) The boundaries are .
c) The boundaries are .
d) The boundaries are .
Step-by-step explanation:
Z-score:
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. The middle 20% from the 80% in the tails.
The middle 20% is between the 50 - (20/2) = 40th percentile and the 50 + (20/2) = 60th percentile:
40th percentile: Z has a pvalue of 0.4, so Z = -0.253.
60th percentile: Z has a pvalue of 0.6, so Z = 0.253.
The boundaries are .
b. The middle 50% from the 50% in the tails.
The middle 50% is between the 50 - (50/2) = 25th percentile and the 50 + (50/2) = 75th percentile:
25th percentile: Z has a pvalue of 0.25, so Z = -0.675.
75th percentile: Z has a pvalue of 0.75, so Z = 0.675.
The boundaries are .
c. The middle 95% from the 5% in the tails.
The middle 95% is between the 50 - (95/2) = 2.5th percentile and the 50 + (95/2) = 97.5th percentile:
2.5th percentile: Z has a pvalue of 0.025, so Z = -1.96.
97.5th percentile: Z has a pvalue of 0.975, so Z = 1.96.
The boundaries are .
d. The middle 99% from the 1% in the tails.
The middle 99% is between the 50 - (99/2) = 0.5th percentile and the 50 + (99/2) = 99.5th percentile:
0.5th percentile: Z has a pvalue of 0.005, so Z = -2.575.
99.5th percentile: Z has a pvalue of 0.995, so Z = 2.575.
The boundaries are .