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.

Hypothesis:

Sample proportion:

Test statistic,

Critical value = 2.33

Since calculated value is greater than critical value, we reject null hypothesis and conclude that more than 65% of Queenslanders agree for the state to re-open for non-essential businesses.

b.

P-value = P(z > 2.436) = 1 - P(z < 2.436) = 1 - 0.9926 = 0.0074

Since p-value is less than 0.01, we reject null hypothesis.

c.

The following condition need to be satisfy:

1. np > 10

2. n(1-p) > 10

Here np = 1500 * 0.68 = 1020 and n(1-p) = 480

Hence both the conditions are satisfy.

d.

Type I error : Rejecting H0, when it is true.

Type II error : Accepting H0, when it is false.

Here we reject null hypothesis, if H0 is actually true then we did type I error.