Question

A sample of size 126 will be drawn from a population with mean 26 and standard deviation 3. Use the TI-84 calculator.

Answer 1

Answer:

1. The probability that x will be more than 25 is 0.6305.

2. The 55th percentile is 26.38.

Step-by-step explanation:

Normal Probability Distribution

Problems of normal distributions can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the z-score 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 p-value, we get the probability that the value of the measure is greater than X.

Mean 26 and standard deviation 3.

This means that $\mu = 26, \sigma = 3$

1 Find the probability that x will be more than 25.

This is 1 subtracted by the p-value of Z when X = 25. So

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

$Z = \frac{25 - 26}{3}$

$Z = -0.333$

$Z = -0.333$ has a p-value of 0.3695.

1 - 0.3695 = 0.6305.

The probability that x will be more than 25 is 0.6305.

2 Find the 55th percentile of x.

This is X when Z has a p-value of 0.55, so X when Z = 0.125.

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

$0.125 = \frac{X - 26}{3}$

$X - 26 = 0.125*3$

$Z = 26.38$

The 55th percentile is 26.38.