Question

The 400-meter race times recorded in the boys track meet was normally distributed with a standard deviation of 3 seconds.If David finished the race in 51.2 seconds with a z-score of -2.6, what is the mean time?

Answer 1

Answer:

The mean time is 59 seconds.

Step-by-step 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:

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

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 question, we have that:

$\sigma = 3$

If David finished the race in 51.2 seconds with a z-score of -2.6, what is the mean time?

This means that when X = 51.2, Z = -2.6. So

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

$-2.6 = \frac{51.2 - \mu}{3}$

$51.2 - \mu = -2.6*3$

$\mu = 51.2 + 2.6*3$

$\mu = 59$

The mean time is 59 seconds.

Answer 2

Answer:

The z score is defined as:

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

If we solve for the mean $\mu$ we got:

$\mu = X -z\sigma$

Replacing we got:

$\mu = 51.2 - (-2.6)*3 = 59$

Step-by-step explanation:

For this case we have the following info given:

$z = -2.6$ represent the z score

$\sigma=3$ represent the population deviation

$x= 51.2$ represent the time for David

The z score is defined as:

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

If we solve for the mean $\mu$ we got:

$\mu = X -z\sigma$

Replacing we got:

$\mu = 51.2 - (-2.6)*3 = 59$