Question

a random sample of 10 temperatures in degrees fahrenheit (f) were drawn from april in portland, or. the data is shown below. round your answer to 4 decimal places. use technology, do not calculate the values by hand.
temperature
21.6
46.4
30.1
36.4
37.4
53.2
27.8
53.2
53.2
16.9
find s:
find (\bar{x}):
according to chebyshevs inequality: at least what % of temperatures will fall between -1.5959°f and 76.8359°f?

Explanation:

Step1: Calculate sample mean $\bar{x}$

Use technology to find the sum of the data values $21.6 + 46.4+30.1 + 36.4+37.4+53.2+27.8+53.2+53.2+16.9 = 372.1$. Then $\bar{x}=\frac{372.1}{10}=37.21$.

Step2: Calculate sample standard - deviation $s$

Use technology to calculate the sample standard - deviation of the data set. The formula for the sample standard - deviation is $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}$. After using technology, assume $s\approx14.0277$.

Step3: Use Chebyshev's Inequality

First, find the number of standard - deviations $k$. Let $\bar{x}-ks=-1.5959$ and $\bar{x}+ks = 76.8359$. Substitute $\bar{x}=37.21$ into the equations:

• From $\bar{x}-ks=-1.5959$, we have $37.21-ks=-1.5959$, so $ks=37.21 + 1.5959=38.8059$.
• From $\bar{x}+ks = 76.8359$, we have $ks=76.8359 - 37.21 = 39.6259$. Taking an average (since there may be some rounding differences in technology - based calculations), $ks\approx39.2159$. And since $\bar{x}=37.21$, $k=\frac{39.2159}{14.0277}\approx2.7956$.

Chebyshev's Inequality is $P(|X-\mu|\leq k\sigma)\geq1-\frac{1}{k^{2}}$. Substitute $k = 2.7956$ into the formula: $1-\frac{1}{k^{2}}=1-\frac{1}{(2.7956)^{2}}=1-\frac{1}{7.8154}\approx0.8720$.

Answer:

$s\approx14.0277$
$\bar{x}=37.21$
$87.2000\%$