The mean price for new homes from a sample of houses is $155,000 with a standard deviation of $10,000. Assume that the data set has a symmetric and bell-shaped distribution.

(a) Between what two values do about 95% of the data fall?

(b) Estimate the percentage of new homes priced between $135,000 and $165,000?

a) 95% of the data falls between $135,000 and $175,000.

b) 81.5% of new homes priced between $135,000 and $165,000.

Step-by-step explanation:

The Empirical Rule states that, for a normally distributed random variable:

68% of the measures are within 1 standard deviation of the mean.

95% of the measures are within 2 standard deviations of the mean.

99.7% of the measures are within 3 standard deviations of the mean.

We also have that:

50% of the measures are below the mean and 50% of the measures are above the mean.

34% of the measures are between 1 standard deviation below the mean and the mean, and 34% of the measures are between the mean and 1 standard deviations above the mean.

47.5% of the measures are between 2 standard deviations below the mean and the mean, and 47.5% of the measures are between the mean and 2 standard deviations above the mean.

49.85% of the measures are between 3 standard deviations below the mean and the mean, and 49.85% of the measures are between the mean and 3 standard deviations above the mean.

In this problem, we have that:

Mean = $155,000.

Standard deviation = $10,000.

(a) Between what two values do about 95% of the data fall?

By the Empirical Rule, 95% of the values fall within 2 standard deviations of the mean.

So

155000 - 2*10000 = 135,000

155000 + 2*10000 = 175,000

95% of the data falls between $135,000 and $175,000.

(b) Estimate the percentage of new homes priced between $135,000 and $165,000?

We have to find how many fall between $135,000 and the mean ($155,000) and how many fall between the mean and $165,000

$135,000 and the mean

$135,000 is two standard deviations below the mean.

By the empirical rule, 47.5% of the measures are between 2 standard deviations below the mean and the mean.

So 47.5% of the measures are between $135,000 and the mean

Mean and $165,000

$165,000 is one standard deviation above the mean.

By the empirical rule, 34% of the measures are between the mean and 1 standard deviations above the mean.

So 34% of the measures are between the mean and $165,000.

$135,000 and $165,000

47.5% + 34% = 81.5% of new homes priced between $135,000 and $165,000.