Question

b) The durations, in minutes, of a sample of 10 telephone calls to a helpline were as follows.
32 15 10 17 32 8 20 38 25 18
The duration of calls to the helpline has a known standard deviation of 7.5 minutes.
i). Stating two necessary assumptions, construct a 95% confidence interval for the mean duration of telephone calls to the helpline.
ii). Hence comment, with justification, on a claim that the mean duration of telephone calls to the helpline is 25 minutes

Answer 1

b) $\bar{x}=\frac{32+15+10+17+32+8+20+38+25+18}{10}=21.5.$

i) Assumptions:

• sample is a simple random sample
• sampling distribution is normally distributed.

$(\bar{x}-z_{crit}\frac{\sigma}{\sqrt{n}},\;\bar{x}+z_{crit}\frac{\sigma}{\sqrt{n}})=(21.5-1.96\frac{7.5}{\sqrt{10}},\;21.5+1.96\frac{7.5}{\sqrt{10}})=(16.85,\;26.15).$

ii) Since the value of 25 min lies in the confidence interval, we can conclude that there is no significant evidence that the mean duration of telephone calls differs from 25 min.