Suppose that scores on a particular test are normally distributed with a mean of 140 and a standard deviation of 20. What is the minimum score needed to be in the top 5% of the scores on the test? Carry your intermediate computations to at least four decimal places, and round your answer to at least one decimal place.

The minimum score needed to be in the top 5% of the scores on the test is 172.9.

Step-by-step explanation:

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

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

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.

In this problem, we have that:

What is the minimum score needed to be in the top 5% of the scores on the test?

The 100-5 = 95th percentile, which is the value of X when Z has a pvalue of 0.95. So it is X when Z = 1.645.

The minimum score needed to be in the top 5% of the scores on the test is 172.9.

the minimum score needed to be in the top 5% of the scores on the test is 173

Step-by-step explanation:

Suppose that scores on the particular test are normally distributed, we would apply the formula for normal distribution which is expressed as

z = (x - µ)/σ

Where

x = scores on the test.

µ = mean score

σ = standard deviation

From the information given,

µ = 140

σ = 20

The probability value for the minimum score needed to be in the top 5% of the scores on the test would be (1 - 5/100) = (1 - 0.05) = 0.95

Looking at the normal distribution table, the z score corresponding to the probability value is 1.65

Therefore,

1.65 = (x - 140)/20

Cross multiplying by 20, it becomes

1.65 × 20 = x - 140

33 = x - 140

x = 33 + 140

x = 173

(b) 240

step-by-step explanation:

first, we use the pythagorean formula using the given side lengths to find x. then we use x to find the side lengths. then we add the side lengths to find the perimeter.

pythagorean theorem formula:

a^2 + b^2 = c^2

(x - 20)^2 + (x - 40)^2 = x^2

x^2 - 40x + 400 + x^2 - 80x + 1600 = x^2

x^2 -120x + 2000 = 0

(x - 100)(x - 20) = 0

x - 100 = 0 or x - 20 = 0

x = 100 or x = 20

we see that the solution x = 20 must be discarded because it will give a negative side length and a side length of 0. the only valid solution is x = 100.

perimeter = sum of the lengths of the sides

perimeter = x + x - 20 + x - 40

perimeter = 3x - 60

replace x with 100.

perimeter = 3(100) - 60

perimeter = 300 - 60

perimeter = 240