Question

Both Questions 4 and 5 are related to the following narrative.  For the purposes of these questions, treat the sample as if it were a simple random sample.

The New York Times/Kaiser Family Foundation survey of Chicago is based on interviews conducted April 21 through May 3 with 1,123 adults who live in Chicago. The sample of telephone exchanges called, both landline and cellphone, was randomly selected by a computer from a complete list of exchanges in Chicago, maintained by Marketing Systems Group of Horsham, Pa. Within each exchange, random digits were added to form a complete telephone number, thus permitting access to listed and unlisted numbers alike. Within each landline household, one adult was designated by a random procedure to be the respondent for the survey. Interviewers made multiple attempts to reach every phone number in the survey, calling back unanswered numbers on different days at different times of both day and evening. In addition to sampling error, the practical difficulties of conducting any survey of the public may introduce other sources of error. Variation in the wording, order and translation of questions, for example, may lead to somewhat different results. In the survey, 62% of residents said they disapproved of Mayor Rahm Emanuel’s job performance.

Question 4

1. Construct a 95% confidence interval for the true population proportion of Chicago residents who disapproved of the mayor’s job performance at that time. To the nearest percentage point, what is the margin of error?  Write a single sentence interpretation of this confidence interval in your own words.

2. Now construct a 99% confidence interval for the above scenario.  Indicate the difference(s) that you see between the confidence interval you calculated in Part (A) and this one.  Then write a single sentence that discusses the implicationsof those differences.

Question 5

1. At the 5% level of significance, is the sampled proportion of residents who disapproved of the mayor significantly different from 0.6? Show the five step process of hypothesis testing and be clear about your work/assumptions/decisions for the test.

2. Why did the researchers contact cell phones and landlines as well as listed and unlisted numbers? Why did they try so hard to reach people who did not answer their calls?

Answer 1

Let the true population proportion of the Chicago residents who disapproved Mayor's job performance at that time be P.

The sample proportion of Chicago residents who disapproved Mayor's job performance = p = 62% = 0.62

Sample size = n = 1123

a) Level of significance = = 1 - 0.95 = 0.05

95% confidence interval for P is given by -   ____ (where M.O.E is the margin of errror)

  

where, is the critical value of z for two tailed test at 0.05 level of significance.

  

Hence, Margin of errror here is 0.03 (or 3%)

= [0.59, 0.65]

Hence, we are 95% confident that the true population proportion of the Chicago's resident who disapproved Mayor's job performance will lie between 59% to 65% (rounding up to nearest percentage)

b) The 99% confidence interval for P will be given by -

where, is the critical value of z for two tailed test at 0.01 level of significance.

  

  

Margin of errror here is 0.04(or 4%)

= [0.58, 0.66]

Hence, we are 99% confident that the true population proportion of the Chicago's resident who disapproved the Mayor's job performance lie between 58% and 66% (rounding up to the nearest percentage)

Comparing the two intervals, we see that 99% confidence interval is wider than the 95% confidence interval (since, margin of errror is greater in case of 99% confidence level) suggesting that the 99% confidence interval is more accurate than the 95% confidence interval.

c) The hypothesis may be framed as -

Null hypothesis : P = 0.60

Alternative hypothesis :

Test statistic is given by -

where Po is the hypothesed value of P under the null hypothesis = 0.60.

Hence, the value of test statistic will be -

  

= 3.45

Now, the critical value of z for two tailed test at 5% level of significance = 1.96

Since, the value of test statistic > critical value of z, hence, the null hypothesis may be rejected at the 0.05 level of significance, hence, the proportion of Chicago residents who disapproved the Mayor's job performance is significantly different from 0.60

d) The effort was made to ensure the randomness of the sample so that the biasness may be minimised to the maximum possible level and the sample may be representative of the population.