## Related questions with answers

A random sample of the weekly supermarket bills for a number of families was observed and recorded in the table given. a. Find the mean bill and the standard deviation of the bills.b. Find unbiased estimates of the mean and variance of the population from which the data was taken.

$\begin{matrix} Bill (\$) & \text{No. of families}\\ \text{70-79.99} & \text{27}\\ \text{80-89.99} & \text{32}\\ \text{90-99.99} & \text{48}\\ \text{100-109.99} & \text{25}\\ \text{110-119.99} & \text{37}\\ \text{120-129.99} & \text{21}\\ \text{130-139.99} & \text{18}\\ \text{140-149.99} & \text{7}\\ \end{matrix}$

Solution

Verified$\textbf{(a)}$ We know that the formula for mean $\bar{x}$ with score value $x_i$, corresponding frequency $f_i$ and total number of items $n$ is,

$\bar{x}=\dfrac{\sum_{i=1}^n x_i\cdot f_i}{\sum_{i=1}^n f_i}$

Also, remember that if data is given in form of classes or groups then the score value will be average of each class.

For the given data we could calculate the mean of the sample using the above formula as follow.

$\begin{align*} \bar{x} &= \dfrac{74.995(27)+84.995(32)+94.995(48)+104.995(25)+114.995(37)+124.995(21)+134.995(18)+144.995(7)}{27+32+48+25+37+21+18+7}\\\\ \bar{x} &= \dfrac{22253.925}{215}\\\\ \bar{x} &= 103.506627907\\\\ \bar{x} &\approx 103.5 \end{align*}$

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