Question

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. |

Answer #1

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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Round your response to at least 3 decimal places.
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(b)
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(b)
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