Question

Queensland is a state with a population of 5.1 million residents. It is reported that Queensland is doing very well in flattening the curve on the number of new cases of the Covid19 infection. Queenslanders are divided on whether or not the state to re-open for nonessential businesses. A survey is conducted over 1,500 residents where they are asked to cast their vote on the issue. 1,020 residents stated ‘yes’ for the state to re-open for non-essential businesses.

a)You were recently hired as a junior statistician working for the Office of the Premier of Queensland. Assist the office in performing a hypothesis test at the 1% level of significance to infer whether more than 65% of Queenslanders agree for the state to re-open for non-essential businesses. 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.

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.

c) 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.

d) Identify which one of these two types of error (Type I or Type II) you could make with the conclusion you made in part a). Briefly explain your selection.

Answer 1

a)

Step 1: HYPOTHESES

Let p be the true proportion of Queenslanders agree for the state to re-open for non-essential businesses.

Null Hypothesis H0: p 0.65

Alternative Hypothesis Ha: p > 0.65

Step 2 : ASSUMPTIONS

np(1-p) = 1500 * 0.65 * (1 - 0.65) = 341.25

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 can be assumed to be less than or equal to 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 = = 0.0123153

Sample proportion, = 1020/1500 =  0.68

Test statistic, z = ( - p) / SE = (0.68 - 0.65)/0.0123153 = 2.44

Step 5: DECISION

Since, test statistic is greater than the critical value, we reject null hypothesis H0.

Step 6: CONCLUSIONS

We conclude that there is significant evidence from the data that the true proportion of Queenslanders agree for the state to re-open for non-essential businesses is greater than 0.65

b)

p-value = P(z > 2.44) = 0.0073

Since, p-value is less than 0.01 significance level, we reject null hypothesis H0 and conclude that there is significant evidence from the data that the true proportion of Queenslanders agree for the state to re-open for non-essential businesses is greater than 0.65.

c)

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

np(1-p) = 1500 * 0.65 * (1 - 0.65) = 341.25

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.

d)

As, we have reject the null hypothesis, we may commit Type I error in case the null hypothesis (true proportion of Queenslanders agree for the state to re-open for non-essential businesses is less than or equal to 0.65) is true.