Q12 · 2021 · Technology-activeSimple familiar6 marks
QUESTION 12 (6 marks)
The life span of batteries manufactured by a company is assumed to be normally distributed with an unknown mean and standard deviation.
A supervisor at the company randomly selects batteries and uses the life spans from this sample to calculate an approximate 95% confidence interval for the population mean of hours.
a)[1 mark]
Determine the mean life span for this sample of batteries.
b)[3 marks]
The standard deviation of the life spans of batteries in this sample is 125 hours. Determine .
c)[2 marks]
Use the result from Question 12b) to explain whether the assumption that the life span of batteries is normally distributed is required to support the supervisor’s calculations.
WORKED SOLUTION
6 marksQCAA guide · typeset solution
ANSWER
hours; ; the normal-population assumption is required.
Worked solution
a) hours.
b) An approximate confidence interval for is
Using the upper value of the 95% confidence interval,
Solving gives , so the sample size used was .
c) Yes, the assumption is required. Because the sample size is less than 30, the distribution of the sample mean cannot be assumed to be normal unless the population is normally distributed.
correctly determines the mean lifetime of the sample
determines an equation in terms of
solves the equation for
expresses sample size rounded to a whole number
makes a suitable statement about whether the assumption is required
discusses the sample size and its relation to a normal distribution
One QCAA sample method typeset for web; criterion wording is adapted from the official marking guide.
View the full QCAA marking guideHow many marks did you earn?
Compare your working with the guide above.