A farmer claims that the mean weight of the melons grown in his farm is at least $1.2$ kg. A rando

Question

A farmer claims that the mean weight of the melons grown in his farm is at least $1.2$ kg. A random sample of $10$ melons is chosen and the weight, $x$ kg, of each melon is recorded. The results are as follows:

$1.037$	$0.914$	$1.019$	$1.234$	$1.110$
$1.417$	$1.008$	$0.846$	$1.105$	$1.331$

Another random sample of $40$ melons is weighed and the results are as follows:

$\sum x=45.738,\qquad\sum x^2=72.576$

By combining the $2$ sets of data, find unbiased estimates of the population mean and population variance of the weight of a melon grown by this farmer.[2]
The farmer decides to change the claim to “the mean weight of the melons grown in his farm is at least $m$ kg”. Based on the combined sample of size $50$, find the set of values of $m$ for which the farmer’s claim is not rejected at the $5\%$ significance level. [3]

Accepted Answer

(i) Mean $1.13518$ kg; variance $0.4200113342\ldots$ kg² (ii) $0\le m<1.285935359\ldots$ kg; upper cutoff approximately $1.29$ kg.

2015 AJC P2 Q10 [Modified]

2015 AJC P2 Q10 [Modified]

Solution:

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