Question

The City Council of a regional area with a population of 15,240 people is committed to do a lockdown of the region if more than 3% of its population contracts the Covid-19 virus. To investigate whether more than 3% of the population have contracted the virus, the health

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authority conducted a random Covid-19 test over 1000 people living in the region; 32 people tested positive to the virus.

a) You were recently hired as a junior statistician working for the Council. Assist the council in performing a hypothesis test at the 1% level of significance to address the problem specified. Display the six steps process (involving drawing the rejection region/s and determining the critical value/s for the decision rule) in performing the test. 6 marks

b) Calculate the p-value of the test above. Display working. State the decision rule should you want to use the p-value method hypothesis testing.

2 marks

c) Now that you have performed the test in part a), what would you suggest to the City Council, i.e. should the regional area be locked down? Yes or no? Why? 1 mark

d) This hypothesis test is conducted on the basis that the sampling distribution of the sample proportion is approximately normally distributed. Specify the required condition to ensure this. Further, check if the condition is satisfied. 1 mark

Answer 1

a)

Step 1: HYPOTHESES

Let p be the true proportion of population contracts the Covid-19 virus.

Null Hypothesis H0: p 0.03

Alternative Hypothesis Ha: p > 0.03

Step 2 : ASSUMPTIONS

np(1-p) = 1000 * 0.03 * (1 - 0.03) = 29.1

Since np(1-p) > 10, the sample size is large enough to approximate the sampling distribution of proportion as normal distribution and conduct a one sample z test. The sample can be assumed to be a random sample and the sample size is less than 5% of the population size.

Step 3: REJECTION REGION

Significance level = 0.01

Z value for 0.01 significance level for right tail test is 2.33. We reject the null hypothesis if test statistic is greater than 2.33

Step 4: CALCULATIONS

Standard error of sample proportion, SE = = 0.00539

Sample proportion, = 32/1000 =  0.032

Test statistic, z = ( - p) / SE = (0.032 - 0.03)/0.00539 = 0.3711

Step 5: DECISION

Since, test statistic is less than the critical value, we fail to reject null hypothesis H0.

Step 6: CONCLUSIONS

We conclude that there is no significant evidence from the data that the true proportion of population contracts the Covid-19 virus is greater than 0.03.

b)

p-value = P(z > 0.3711) = 0.3553

Since, p-value is greater than 0.01 significance level, we fail to reject null hypothesis H0 and conclude that there is no significant evidence from the data that the true proportion of population contracts the Covid-19 virus is greater than 0.03.

c)

We should suggest to the City Council, that there is no sufficient evidence that the proportion of population contracts the Covid-19 virus is greater than 0.03 and the regional area should not be locked down.

d)

The required condition to ensure that the sampling distribution of the sample proportion is approximately normally distributed.

np(1-p) = 1000 * 0.03 * (1 - 0.03) = 29.1

Since np(1-p) > 10, the sample size is large enough to approximate the sampling distribution of proportion as normal distribution and conduct a one sample z test.