Question

the following data represent the number of people aged 25 to 64 years covered by health insurance (private or government) in 2018. approximate the mean and standard deviation for age.

age	number (millions)
35 - 44	37.4
45 - 54	36.5
55 - 64	29.1

\\(mu = 45.20\\) (type an integer or decimal rounded to two decimal places as needed.)
\\(sigma = \square\\) (type an integer or decimal rounded to two decimal places as needed.)

Explanation:

Step1: Find the midpoint of each age group

For 25 - 34: Midpoint $x_1=\frac{25 + 34}{2}=29.5$
For 35 - 44: Midpoint $x_2=\frac{35+44}{2}=39.5$
For 45 - 54: Midpoint $x_3=\frac{45 + 54}{2}=49.5$
For 55 - 64: Midpoint $x_4=\frac{55+64}{2}=59.5$

Step2: List the frequencies ($f_i$)

$f_1 = 22.9$, $f_2=37.4$, $f_3 = 36.5$, $f_4=29.1$

Step3: Calculate the total frequency ($N$)

$N=\sum f_i=22.9 + 37.4+36.5 + 29.1=125.9$

Step4: Calculate the mean ($\mu$) (already given as 45.20, but let's verify)

$\mu=\frac{\sum f_ix_i}{N}$
$\sum f_ix_i=22.9\times29.5+37.4\times39.5 + 36.5\times49.5+29.1\times59.5$
$22.9\times29.5 = 675.55$
$37.4\times39.5=1477.3$
$36.5\times49.5 = 1806.75$
$29.1\times59.5=1731.45$
$\sum f_ix_i=675.55 + 1477.3+1806.75+1731.45=5691.05$
$\mu=\frac{5691.05}{125.9}\approx45.20$ (matches the given mean)

Step5: Calculate the variance ($\sigma^2$)

$\sigma^2=\frac{\sum f_i(x_i-\mu)^2}{N}$
First, calculate $(x_i - \mu)^2$ for each group:
For $x_1 = 29.5$: $(29.5 - 45.20)^2=(- 15.7)^2 = 246.49$
For $x_2=39.5$: $(39.5 - 45.20)^2=(-5.7)^2 = 32.49$
For $x_3 = 49.5$: $(49.5 - 45.20)^2=(4.3)^2 = 18.49$
For $x_4=59.5$: $(59.5 - 45.20)^2=(14.3)^2 = 204.49$

Then, calculate $f_i(x_i - \mu)^2$ for each group:
$f_1(x_1-\mu)^2=22.9\times246.49 = 22.9\times246.49\approx5644.621$
$f_2(x_2-\mu)^2=37.4\times32.49=37.4\times32.49\approx1215.126$
$f_3(x_3-\mu)^2=36.5\times18.49=36.5\times18.49\approx674.885$
$f_4(x_4-\mu)^2=29.1\times204.49=29.1\times204.49\approx5950.659$

Sum these up: $\sum f_i(x_i - \mu)^2=5644.621+1215.126 + 674.885+5950.659=13485.291$

Then, $\sigma^2=\frac{13485.291}{125.9}\approx107.11$

Step6: Calculate the standard deviation ($\sigma$)

$\sigma=\sqrt{\sigma^2}=\sqrt{107.11}\approx10.35$

Answer:

$\sigma\approx\boldsymbol{10.35}$