Confidence Intervals Exercises

1. You are interested in the mean duration of a spell of unemployment for currently unemployed women in a particular city. It is known that the unemployment duration of women is normally distributed with variance [math]129.6[/math]. The units of measurement for the variance are therefore months squared. You draw a random sample of [math]20[/math] unemployed women, and they have an average unemployment duration of [math]14.7[/math] months. Obtain a [math]98\%[/math] confidence interval for the population mean unemployment duration for women.

2. A simple random sample of [math]15[/math] pupils attending a certain school is found to have an average IQ of [math]107.3[/math] with a sample variance of [math]32.5[/math].

   1. Calculate a [math]95\%[/math] confidence interval for the unknown population mean IQ, stating any assumptions you need to make. Interpret this interval.

   2. Explain whether you would be happy with a parent’s claim that the average IQ at the school is [math]113[/math].

3. An internet service provider is investigating the length of time its subscribers are connected to its site, at any one visit. Having no prior information about this, it obtains three random samples of these times, measured in minutes. The first has sample size 25, the second sample size 100 and the third 250. The sample information is given in the table below:

   	[math]n[/math]	[math]\bar{x}[/math]	[math]s^{2}[/math]
   Sample 1	[math]25[/math]	[math]9.8607[/math]	[math]2.1320[/math]
   Sample 2	[math]100[/math]	[math]9.8270[/math]	[math]2.1643[/math]
   Sample 3	[math]250[/math]	[math]9.9778[/math]	[math]2.0025[/math]

   1. Calculate a [math]95\%[/math] confidence interval for each sample: state any assumptions you make.

   2. Do the confidence intervals get narrower as the sample size increases? Why would you expect this?

4. In an opinion poll based on [math]100[/math] interviews, [math]34[/math] people say they are not satisfied with the level of local Council services. Find a [math]99\%[/math] confidence interval for the true proportion of people who are not satisfied with local Council services.

5. Explain the difference between

   1. an interval estimator and an interval estimate;

   2. an interval estimate and a confidence interval.

6. Which of these interpretations of a [math]95\%[/math] confidence interval [math]\left[c_{L},c_{U}\right] [/math] for a population mean are valid, and why?

   1. [math]\mu [/math] lies in the interval [math]\left[ c_{L},c_{U}\right] [/math] with probability [math]0.95[/math];

   2. in repeated sampling, approximately [math]95\%[/math] of confidence intervals will contain [math]\mu ;[/math]

   3. [math]\mu [/math] lies in the interval [math]\left[ C_{L},C_{U}\right] [/math] with probability [math]0.95[/math];

   4. the confidence interval [math]\left[ c_{L},c_{U}\right] [/math] contains a point estimate of [math]\mu [/math] and an allowance for sampling variability;

   5. [math]\left[ c_{L},c_{U}\right] [/math] displays the likely range of values of [math]\mu[/math].

   6. [math]\left[ c_{L},c_{U}\right] [/math] shows how precise the estimator of [math]\mu[/math] is expected to be.