Question

A manufacturer of orthopaedic equipment believes that specialised hospitals that have units dedicated to traumatology are a large part of the market in the USA, because these hospitals tend to buy more orthopedic equipment.

The manufacturer collected a sample and the data shows that out of a total of 326 specialised hospitals in 8 states in the USA, 173 have traumatology units.

1. We wish to test if there is sufficient evidence at the 1% significance level to suggest that the majority of the specialised hospitals in the USA have traumatology units.

   1. What is the hypothesised value of the population parameter?

   2. Perform an appropriate hypothesis test. Use the critical value approach and show all working

2. Comment on the manufacturer’s method of choosing a sample.

Answer 1

a)We are asked to test the hypothesis at 1% level that the majority of the specialised hospitals in the USA have traumatology units and are given the following values.

n=326(total no. of specialised hospitals), x=173(no. of hospitals that have traumatology units)

(sample proportion of hospitals)

i) P=0.5.

Here P is the hypothesised population parameter.

=>Q=1-P=1-0.5=0.5

ii)

1)Null hypothesis

i.e., 50% of the specialised hospitals in the USA have traumatology units.

2)Alternate hypothesis

(one-tailed test)

i.e., majority of the specialised hospitals in thr USA have traumatology units.

3) Rejection Region

We reject null hypothesis, H₀ if (which is the critical value) at 1% level of significance.

The shaded region here represents the rejection region at 1% level of significance for a one-tailed test.

4) Given alpha level is 1%=0.01

5) Test statistic under H₀

6) Conclusion

at 1% level of significance for a one-tailed test

Since => null hypothesis is accepted.

Hence we conclude that 50% of the specialised hospitals have traumatology units i.e., the manufacturer's claim of majority of the specialised hospitals in the USA have traumatology units is wrong.

b) The manufacturer seems to have used cluster sampling method to choose a sample because in cluster sampling method we divide the data into clusters and then select one or a few clusters and then sample everyone from the chosen subset.

In our case all the states in the USA form the population where each state is a cluster. From which the manufacturer selected 8 states or clusters in cluster sampling notation and then sampled every hospital in the 8 clusters which gave us a total of 326 hospitals from these 8 states.