Question 4
The weekly amount of money a company spends on repairs is normally distributed, with a mean of and a standard deviation of .
Given that $P(Z\le-2.5)=0.0062$ and $P(Z\gt1)=0.1587$, where $Z$ is a standard normal random variable, determine the probability that the weekly repair costs will be between and .
Not quite. 0.6525 is $0.5-0.0062+0.1587$. The first part, $0.5-0.0062$, is right for up to the mean. But 0.1587 is the tail above , not the area from the mean to . That piece is $0.5-0.1587=0.3413$. In 2022, 11.7% of Queensland students chose (A).
Not quite. 0.6587 is $0.5+0.1587$. That adds the tail above instead of taking it away, and it ignores the tail below . In 2022, 17.7% of Queensland students chose (B).
That’s it. Step through the hints below to see why. In 2022, 57.9% of Queensland students chose (C).
Close. 0.8413 is $1-0.1587$, which is $P(Z\lt1)$. You’ve removed the top tail but not the bottom one. Costs below also need to come off: $0.8413-0.0062=0.8351$. In 2022, 11.6% of Queensland students chose (D).
This is the question’s interval, to . The two grey tails are exactly the areas you were given, so take both away from 1.
Both bounds are at the mean, so there’s no width between them and no area. The probability of a single exact cost is 0.
The orange area is the probability between the two bounds. The grey tails are what you take away from the total area of 1.
Turn each cost into a $z$-score with $z=\dfrac{x-\mu}{\sigma}$.
$\dfrac{950-1200}{100}=-2.5$$\dfrac{1300-1200}{100}=1$
$P(950\lt X\lt1300)$$=P(-2.5\lt Z\lt1)$
Sketch it. The area you want is the middle. The two tails are exactly the probabilities you were given.
Left tail: $P(Z\le-2.5)=0.0062$
Right tail: $P(Z\gt1)=0.1587$
The whole area under the curve is 1, so take both tails away.
$P(-2.5\lt Z\lt1)$$=1-0.0062-0.1587$$=0.8351$
$P(950\lt X\lt1300)$$=P(-2.5\lt Z\lt1)$
$=1-P(Z\le-2.5)-P(Z\gt1)$
$=1-0.0062-0.1587$$=0.8351$
How Queensland students answered in 2022
From the QCAA Mathematical Methods subject report for 2022. The percentages can add to less than 100 because some students didn’t answer.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2022 QCAA Mathematical Methods external assessment, © State of Queensland (QCAA) 2022, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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