A nasi lemak stall holder uses fresh chicken wings as an ingredient for fried chicken wings. Based o

Question

A nasi lemak stall holder uses fresh chicken wings as an ingredient for fried chicken wings. Based on his past years records, his mean daily profit was $\$535$. With the recent lack of fresh chicken supply, the stall holder substituted fresh chicken wings with frozen chicken wings as the ingredient. His wife was hesitant to the change and claimed that the mean daily profit will decrease. To test his wife’s claim, the stall holder takes a random sample of $45$ days and recorded the daily profits, $\$x$.

State appropriate hypotheses to test the wife’s claim and define any symbols that you use.[2]
State, with a reason, whether it is necessary to assume that his past years records of daily profits are normally distributed for the test to be valid.[1]
Based on the past years records, it is assumed that the population variance of the daily profit is $2591$. If the test shows that there is sufficient evidence that the wife’s claim is accepted at $5\%$ level of significance, determine the set of possible values of $\bar{x}$, the mean daily profit in the $45$ days.[2]
The stall holder found that $\bar{x}=\$520$ and suspects that the population variance of $\mathrm{2591}$ may be incorrect. Hence, he decided to use the sample variance value of $2008$ to test his wife’s claim. State the conclusion of the test, showing your workings clearly.[3]
State the largest significance level that the stall holder should use so that the conclusion in (iv) will be different. Leave your answer in $2$ decimal places.[1]

The stall holder now suspects that the mean daily profit does not differ from $\$535$, even if frozen chicken wings are used instead of fresh chicken wings. To test his claim, the stall holder decides to increase the number of randomly selected days, $n$, to record his daily profit.

It is given that $\bar{x}=526$ and the population variance is assumed to be $\mathrm{2591}$. Determine the greatest value of $n$, so that the conclusion of the test shows that there is no reason to reject the null hypothesis at $8\%$ level of significance.[4]

Accepted Answer

(i) To test ${H_0 : \mu = 535}$ against ${H_1 : \mu < 535}$ where ${\mu}$ represents the population mean daily profit of the nasi lemak stall holder. (ii) It is not necessary to assume that the daily profits are normally distributed for the test to be valid since ${n = 45}$ is large, Central Limit Theorem applies. (iii) ${\{\bar{x} : 0 < \bar{x} < 523\}}$ (3 s.f.) (iv) ${p\text{ }-\text{ value} = 0.013195 < 0.05 \ \therefore}$ Reject ${H_0}$. There is sufficient evidence at the ${5\%}$ level of significance to conclude that the population mean daily profit is less than ${\$535}$. (v) ${1.31\%}$ (vi) Greatest value of ${n}$ is ${98}$.

2022 ACJC P2 Q10

2022 ACJC P2 Q10

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