Question

the selling prices for homes in a certain community are approximately normally distributed with a mean of $321,000 and a standard deviation of $38,000. estimate the percentage of homes in this community with selling prices

(a) between $245,000 and $397,000.

%

(b) above $435,000.

%

(c) below $245,000.

%

(d) between $245,000 and $435,000.

%

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Explanation:

Part (a)

Step1: Calculate z - scores

For \(x = 245000\), the z - score formula is \(z=\frac{x-\mu}{\sigma}\), where \(\mu = 321000\) and \(\sigma=38000\).
\(z_1=\frac{245000 - 321000}{38000}=\frac{- 76000}{38000}=- 2\)
For \(x = 397000\),
\(z_2=\frac{397000 - 321000}{38000}=\frac{76000}{38000}=2\)

Step2: Use empirical rule

For a normal distribution, the percentage of data between \(z=- 2\) and \(z = 2\) is approximately \(95\%\) (from the empirical rule: about \(95\%\) of data lies within \(2\) standard deviations of the mean).

Step1: Calculate z - score

For \(x = 435000\),
\(z=\frac{435000 - 321000}{38000}=\frac{114000}{38000}=3\)

Step2: Use empirical rule

The empirical rule states that about \(99.7\%\) of data lies within \(3\) standard deviations of the mean. So the percentage of data above \(z = 3\) is \(\frac{1 - 0.997}{2}=0.15\%\) (since the normal distribution is symmetric, the area in the two tails beyond \(z = 3\) and \(z=-3\) is \(1 - 0.997 = 0.003\), and we want the area in the upper tail).

Step1: Recall z - score from part (a)

For \(x = 245000\), \(z=-2\)

Step2: Use empirical rule

From the empirical rule, the percentage of data below \(z=-2\) is the same as the percentage above \(z = 2\). Since the total area outside \(z=-2\) and \(z = 2\) is \(1 - 0.95=0.05\), the area below \(z=-2\) is \(\frac{0.05}{2}=2.5\%\)

Answer:

\(95\)

Part (b)