2) It is believed that 12.5% of adults have skin allergies. Suppose we take a sample of 135 individuals from the population of adults. Answer the following questions.

(a)What is the population mean of the proportion of adults in the sample who have skin allergies?(2 marks)

16.875

0.125

14.766

12.5

135

(b)What is the population standard deviation of the proportion of adults in the sample who have skin allergies?(2 marks)

14.766

0.0008101852

3.843

0.02846375

16.875

(c)What is the distribution of the proportion of adults in the sample who have skin allergies?(2 marks)

N(16.875, 14.766)

N(0.125, 0.0008101852)

B(12.5, 14.766)

N(0.125, 0.02846375)

B(0.125, 4.765)

(d)What is the probability that the proportion of adults in the sample who have skin allergies is less than 0.11?(2 marks)

0.6257

0.2981

0.7019

0.3743

0.5981

(e)What is the probability that the proportion of adults in the sample who have skin allergies is at least 0.12?(2 marks)

0.6712

0.5714

0.3288

0.4286

0.6923

(f)What is the probability that the proportion of adults in the sample who have skin allergies is between 0.13 and 0.145?(2 marks)

0.1866

0.3754

0.6354

0.0982

0.1091

3)A research team believes that the heights of Australian adult males are normally distributed. A simple random sample of 19 males yielded a mean of 174 cm and a standard deviation of 6 cm. Letdenote the mean height for Australian adult males. Suppose the research team wants to test, at the 2.5% level of significance, whether the mean height for Australian adult males is less than 175 cm. Suppose the assumptions to this hypothesis test are satisfied.

(a)What are the null and alternative hypotheses, and level of significance, for this hypothesis test?(2 marks)

H0: = 175 vs Ha:< 175 ( = 0.025)

H0: = 174 vs Ha:> 174 ( = 0.025)

H0: = 175 vs Ha: 175 ( = 0.025)

H0: = 175 vs Ha:> 175 ( = 0.025)

H0: = 174 vs Ha:< 174 ( = 0.025)

(b)What is the distribution of the test statistic for this hypothesis test?(2 marks)

N(174, 36)

t18

N(175, 36)

N(0, 1)

t19

(c)What is the decision rule for this hypothesis test?(2 marks)

We rejectH0if t 2.0930

We rejectH0if t 2.0930

We rejectH0if z 1.9600

We rejectH0if t 2.1009

We rejectH0if t 2.1009

We rejectH0if z 1.9600

(d)What is the observed test statistic for this hypothesis test?(2 marks)

0.7265

2.5823

2.5823

0.7265

0.1211

0.1211

(e)What is the correct conclusion for this hypothesis test?(2 marks)

Since the observed test statistic is greater than the critical value we rejectH0at the 2.5% level of significance.

Since the observed test statistic is less than the critical value we do not rejectH0at the 2.5% level of significance.

Since the observed test statistic is less than the critical value we rejectH0at the 2.5% level of significance.

Since the observed test statistic is greater than the critical value we do not rejectH0at the 2.5% level of significance.