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Math  /  Data & Statistics

QuestionThe table shows information about the weekly earnings of 20 peop who work in a shop. \begin{tabular}{|c|c|} \hline Weekly earnings ( fx\mathbf{f} \boldsymbol{x} ) & Frequency \\ \hline 150<x⩽250150<x \leqslant 250 & 1 \\ \hline 250<x⩽350250<x \leqslant 350 & 11 \\ \hline 350<x⩽450350<x \leqslant 450 & 5 \\ \hline 450<x⩽550450<x \leqslant 550 & 0 \\ \hline 550<x⩽650550<x \leqslant 650 & 3 \\ \hline \end{tabular}
Work out an estimate for the mean of the weekly earnings.

Studdy Solution

STEP 1

What is this asking? Find the average weekly earnings based on the data in the table. Watch out! Don't just add up the frequency numbers and divide!
We need to consider the earnings ranges.

STEP 2

1. Find Midpoints
2. Calculate Total Earnings
3. Divide for the Mean

STEP 3

To estimate the mean, we first need to find the **midpoint** of each earnings range.
This midpoint represents the average earnings within that range.
For the first range, $150<x⩽$250\$150 < x \leqslant \$250, the midpoint is $150+$2502=$4002=$200\frac{\$150 + \$250}{2} = \frac{\$400}{2} = \$200.

STEP 4

Let's do this for all the ranges!
For $250<x⩽$350\$250 < x \leqslant \$350, the midpoint is $250+$3502=$6002=$300\frac{\$250 + \$350}{2} = \frac{\$600}{2} = \$300.
For $350<x⩽$450\$350 < x \leqslant \$450, it's $350+$4502=$8002=$400\frac{\$350 + \$450}{2} = \frac{\$800}{2} = \$400.
For $450<x⩽$550\$450 < x \leqslant \$550, it's $450+$5502=$10002=$500\frac{\$450 + \$550}{2} = \frac{\$1000}{2} = \$500.
Lastly, for $550<x⩽$650\$550 < x \leqslant \$650, the midpoint is $550+$6502=$12002=$600\frac{\$550 + \$650}{2} = \frac{\$1200}{2} = \$600.

STEP 5

Now, we **multiply** each midpoint by its corresponding **frequency** to estimate the total earnings for each range.
This is like saying, "If one person earns \$200, then the total earnings for that range is \(1 \cdot \$200 = \$200\)."

STEP 6

Let's continue!
For the range with a midpoint of \$300 and a frequency of 11, the total earnings are \(11 \cdot \$300 = \$3300\).
For the \$400 midpoint and frequency 5, it's \(5 \cdot \$400 = \$2000\).
For the \$500 midpoint and frequency 0, it's \(0 \cdot \$500 = \$0\).
And for the \$600 midpoint and frequency 3, it's \(3 \cdot \$600 = \$1800\).

STEP 7

To get the **overall estimated total earnings**, we add up the total earnings from each range: $200+$3300+$2000+$0+$1800=$7300\$200 + \$3300 + \$2000 + \$0 + \$1800 = \$7300.

STEP 8

Finally, to find the **estimated mean**, we divide the overall estimated total earnings by the **total number of people**, which is 20.
So, the estimated mean weekly earnings is $730020=$365\frac{\$7300}{20} = \$365.

STEP 9

The estimated mean of the weekly earnings is $365\$365.

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