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Mathematics · 2023 · 5 marks
CBSE 2023 · Region 6 · Set 1 · Q35
250 apples of a box were weighed and the distribution of masses of the apples is given in the following table: (i) Find the value of x and the mean…
$\displaystyle 250$ apples of a box were weighed and the distribution of masses of the apples is given in the following table :
| Mass (in grams) | $\displaystyle 80$-$\displaystyle 100$ | $\displaystyle 100$-$\displaystyle 120$ | $\displaystyle 120$-$\displaystyle 140$ | $\displaystyle 140$-$\displaystyle 160$ | $\displaystyle 160$-$\displaystyle 180$ |
| Number of apples | $\displaystyle 20$ | $\displaystyle 60$ | $\displaystyle 70$ | $\displaystyle x$ | $\displaystyle 60$ |
(i)
Find the value of $\displaystyle x$ and the mean mass of the apples.
(ii)
Find the modal mass of the apples.
Marking-scheme solution
(i)
\(\displaystyle 20+60+70+\mathrm{x}+60=250\)
\[x=250-210=40
\]
| Mass | $\displaystyle 80$-$\displaystyle 100$ | $\displaystyle 100$-$\displaystyle 120$ | $\displaystyle 120$-$\displaystyle 140$ | $\displaystyle 140$-$\displaystyle 160$ | $\displaystyle 160$-$\displaystyle 180$ | Total |
| No. of apples \(\displaystyle \mathrm{f}_{\mathrm{i}}\) | $\displaystyle 20$ | $\displaystyle 60$ | $\displaystyle 70$ | \(\displaystyle \mathrm{x}=40\) | $\displaystyle 60$ | $\displaystyle 250$ |
| \(\displaystyle \mathrm{x}_{\mathrm{i}}\) | $\displaystyle 90$ | $\displaystyle 110$ | $\displaystyle 130$ | $\displaystyle 150$ | $\displaystyle 170$ | |
| \(\displaystyle \mathrm{x}_{\mathrm{i}} \mathrm{f}_{\mathrm{i}}\) | $\displaystyle 1800$ | $\displaystyle 6600$ | $\displaystyle 9100$ | $\displaystyle 6000$ | $\displaystyle 10200$ | $\displaystyle 33700$ |
Mean mass \(\displaystyle =\frac{33700}{250}=134 \cdot 8\)
Mean mass \(\displaystyle =134.8 \mathrm{~g}\)
(ii) Modal class = $\displaystyle 120$-$\displaystyle 140$
\[\begin{aligned}
\text { Mode } & =120+\frac{(70-60)}{(140-60-40)} \times 20 \\
& =125
\end{aligned}
\]
Hence modal mass \(\displaystyle =125 \mathrm{gm}\)
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