Question

# Test the null hypothesis H0:p=0.5against the alternative hypothesis HA:p<0.5, when 92 individuals in a random sample...

Test the null hypothesis H0:p=0.5against the alternative hypothesis HA:p<0.5, when 92 individuals in a random sample of 219 have a characteristic of interest.

Proportions are very sensitive to round-off error. Please ensure that you attempt to round p^as little as possible.

a) Calculate the value of the z test statistic, for testing the null hypothesis that the population proportion is 0.5.

Round your response to at least 3 decimal places.

b) The p-value falls within which one of the following ranges:

 p-value > 0.50 0.10 < p-value < 0.50 0.05 < p-value < 0.10 p-value < 0.05

c) What conclusion can be made, at the 5% level of significance?

 There is insufficient evidence to reject the null hypothesis, and therefore no significant evidence that the population proportion is not 0.5. There is sufficient evidence to reject the null hypothesis, in favour of the alternative that the population proportion is less than 0.5.

Solution:

Part a

Test statistic formula for this test is given as below:

Z = (p̂ - p)/sqrt(pq/n)

Where, p̂ = Sample proportion, p is population proportion, q = 1 - p, and n is sample size

x = number of items of interest = 92

n = sample size = 219

p̂ = x/n = 92/219 = 0.420091324

p = 0.5

q = 1 - p = 0.5

Z = (p̂ - p)/sqrt(pq/n)

Z = (0.420091324 – 0.5)/sqrt(0.5*0.5/219)

Z = -2.3651

Test statistic = Z = -2.365

Part b

P-value = 0.0090

(by using z-table)

p-value < 0.05

Part c

P-value < 0.05

So, we reject the null hypothesis at 5% level of significance

Conclusion:

There is sufficient evidence to reject the null hypothesis, in favour of the alternative that the population proportion is less than 0.5.

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