Question

in a survey of 600 homeowners with high speed internet the average monthly cost of high spped internet was 64.20 with a standard deviation 11.77 assume the plan costs to be approximately bell shaped . estimate the number of plans cost between 40.66 and 87.44 and 52.44 and 75.97

Answer 1

Answer:

572 plans cost between 40.66 and 87.44

410 plans cost between 52.44 and 75.97

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 problem, we have that:

$\mu = 64.20, \sigma = 11.71$

Estimate the number of plans cost between 40.66 and 87.44

The first step is finding the percentage of plans between 40.66 and 87.44, which is the pvalue of Z when X = 87.44 subtracted by the pvalue of Z when X = 40.66

X = 87.44

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

$Z = \frac{87.44 - 64.20}{11.77}$

$Z = 1.97$

$Z = 1.97$ has a pvalue of 0.9756

X = 40.66

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

$Z = \frac{40.66 - 64.20}{11.77}$

$Z = -2$

$Z = -2$ has a pvalue of 0.0228

0.9756 - 0.0228 = 0.9528

95.28% of the plans cost between 40.66 and 87.44.

Out of 600

0.9528*600 = 572

572 plans cost between 40.66 and 87.44

Estimate the number of plans cost between 52.44 and 75.97

The first step is finding the percentage of plans between 52.44 and 75.97, which is the pvalue of Z when X = 75.97 subtracted by the pvalue of Z when X = 52.44

X = 75.97

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

$Z = \frac{75.97 - 64.20}{11.77}$

$Z = 1$

$Z = 1$ has a pvalue of 0.8413

X = 52.44

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

$Z = \frac{52.44 - 64.20}{11.77}$

$Z = -1$

$Z = -1$ has a pvalue of 0.1587

0.8413 - 0.1587 = 0.6826

68.26% of the plans cost between 52.44 and 75.97

Out of 600

0.6826*600 = 410

410 plans cost between 52.44 and 75.97