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

Answer:

a) $Z = -0.8$

b) $Z = 2.4$

c) Mary's score was 241.25.

Step-by-step 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:

$Z = \frac{X - \mu}{\sigma}$

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

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{190 - 210}{25}$

$Z = -0.8$

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

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{270 - 210}{25}$

$Z = 2.4$

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

$Z = \frac{X - \mu}{\sigma}$

$1.25 = \frac{X - 210}{25}$

$X - 210 = 25*1.25$

$X = 241.25$

Mary's score was 241.25.

Answer 2

Answer:

Step-by-step 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