Question

In a recent contest, the mean score was 210 and the standard deviation was 25. a) Find the z-score of John who scored 190 b) Find the z-score of Bill who scored 270 c) If Mary had a score of 1.25, what was Mary’s score?

Answer 1

Explanation:

The formula for normal distribution is expressed as

z = (x - µ)/σ

Where

x = scores in the contest.

µ = mean score.

σ = standard deviation

From the information given,

µ = 210

σ = 25

a) The z-score of John who scored 190

z = (190 - 210)/25 = - 20/25 = - 0.8

b) The z-score of Bill who scored 270

z = (270 - 210)/25 = 60/25 = 2.4

c) If Mary had a score of 1.25, what was Mary’s score,

1.25 = (x - 210)/25

Cross multiplying,

25 × 1.25 = x - 210

x = 31.25 + 210 = 241.25

Mary's score is 241.25

Answer 2

a)
`Z = -0.8`

b)
`Z = 2.4`

c) Mary's score was 241.25.

Explanation:

Problems of normally distributed samples can be 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.

In this problem, we have that:

`\mu = 210, \sigma = 25`

a) Find the z-score of John who scored 190

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

`Z = (190 - 210)/(25)`

`Z = -0.8`

b) Find the z-score of Bill who scored 270

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

`Z = (270 - 210)/(25)`

`Z = 2.4`

c) If Mary had a score of 1.25, what was Mary’s score?

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

`1.25 = (X - 210)/(25)`

`X - 210 = 25*1.25`

`X = 241.25`

Mary's score was 241.25.