S3 June 2013 Q4

EdexcelOld spec14 marksIncludes hypothesis testingChi-Squared Tests

4. Customers at a post office are timed to see how long they wait until being served at the counter. A random sample of 50 customers is chosen and their waiting times, \(x\) minutes, are summarised in Table 1.

Waiting time in minutes (\(x\))Frequency
0–38
3–512
5–613
6–89
8–128

Table 1

(a) Show that an estimate of \(\bar{x} = 5.49\) and an estimate of \(s_x^2 = 6.88\) (3)

The post office manager believes that the customers’ waiting times can be modelled by a normal distribution.
Assuming the data is normally distributed, she calculates the expected frequencies for these data and some of these frequencies are shown in Table 2.

Waiting Time\(x \lt 3\)3–55–66–8\(x \gt 8\)
Expected Frequency8.5612.737.56\(a\)\(b\)

Table 2

(b) Find the value of \(a\) and the value of \(b\). (3)
(c) Test, at the 5% level of significance, the manager’s belief. State your hypotheses clearly. (8)
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