1. Z scores and the Area under the Normal Curve

a. Find the proportion of observations (area under the curve) from a standard normal distribution that satisfies each of the following statements:

i. Z > -0.63

ii. Z < 2.07

iii. -1.23 <Z< 1.46

b. Find the value of the Z score that satisfies each of the following conditions:

i. The value of Z with 25% of observations falling below it;

ii. The value of Z with 34% of observations falling above it.

c. Suppose respondents in a survey were asked about how long they think a terminally ill patient must wait after diagnosis before they should be allowed to end their life with medical assistance (in months). The variable that represents waiting time (in months) from receiving their diagnosis to being able to end their life with medical assistance has an approximate normal distribution, with a mean of 18 months and a standard deviation of 6 months.

i. What proportion of patients would have to wait at least 24 months?

ii. What proportion of patients would have a waiting period take between 4 and 12 months?

iii. What is the waiting time associated with 15% of patients falling below it?

d. The distribution of Australian voters with a relative who has been diagnosed with a terminal illness is skewed to the right, with a μ=1.6 and σ = 0.4. These values of the population are known to you as the researcher who takes a sample of voters from the population of Australian voters in order to estimate the mean number of relatives who have been diagnosed with a terminal illness for each Australian voter. If you select a random sample of 345 voters:

i. Describe the sampling distribution of the sample mean. In other words, write down the appropriate probability distribution and the mean and standard error of this distribution.

ii. Find the probability that your sample mean is greater than 2

2. Confidence Intervals and Hypothesis Tests

a. A sociologist uses a random sample to survey 833 women. She finds that the mean amount of confidence in the current Australian Minister for Health is 4 (measured on a scale of 1-10 where 1 is the lowest amount of confidence and 10 is the highest amount of confidence). The sample variance (s2) of the variable “confidence in the Health Minister” was 1.8.

i. Construct a 95% confidence interval for the mean level of confidence in the Health Minister. Interpret this interval

ii. Another sociologist says that he knows that in the population from which this sample is drawn, the mean level of confidence in the Health Minister is actually 6. Carry out an appropriate statistical test to examine the null hypothesis. Interpret your results.

b. A random sample of 167 Queensland residents was asked what the most important priority is for the Australian criminal justice system. “Bringing back the use of the death penalty” was the first choice for 42 people.

i. Construct a 95% confidence interval for the proportion of people who picked “bringing back the use of the death penalty” as their first choice. Interpret this.

ii. Using the data above, test the null hypothesis that the proportion of people who believe “bringing back the use of the death penalty” is the highest priority is 0.31. Make sure you use the appropriate estimate of the population proportion (as specified under the null hypothesis) in constructing the standard error. Interpret your findings.

3. Confidence Intervals, Hypothesis Tests, and Chi Square

a. A random sample of 45 Brisbane adults report that they perceive that the average cost per week of keeping a convicted murderer in prison (as opposed to using the death penalty) is $3500, with a standard deviation of $990.

i. Identify and describe the correct probability distribution and explain why this is appropriate

ii. Construct a 99% confidence interval to estimate the population mean for “perceived cost of keeping a convicted murderer in prison” for all Brisbane adults as a whole. Fully interpret the interval

iii. Assume that we know that in the Queensland population, the perceived amount of money it costs to keep a convicted murder in prison has an average of $2900. Is the Brisbane sample significantly different from the Queensland average? Use an alpha of 0.01

b. Social science researchers are trying to understand what types of attitudes are related to views about abortion. A random sample of Australians was surveyed about abortion attitudes and political preferences. Below is a contingency table showing the distribution of respondent’s preferred party (ALP or LIB/NAT) and whether they believe that women should have the right to access abortion services (YES or NO).

i. Is there a statistically significant relationship between these variables? You can use the table below for your working.

ii. Compute the column percentages to determine the pattern of the relationship. Which group is more likely to think that abortion should not be allowed?