Inferences Based on a Single Sample Chapter 5 Inferences Based on a Single Sample

Identifying and estimating a target parameter Section 5.1 Identifying and estimating a target parameter

Target Parameter The UNKNOWN population parameter µ σ p

Point Estimate The Single Statistic from our sample data that is our best estimate for our target parameter.  

Interval Estimate A range of values around our point estimate that is likely to contain our target parameter AKA – Confidence Interval

General Formula for Confidence Interval Point Estimate ± (critical value) (s of distr. of point estimate) [p.e. – c.v(s of p.e), p.e + c.v(s of p.e.) ]

   

 

  0.475 0.475

Confidence Level 100(1-α) α α/2 z α/2 90% 0.10 0.05 1.645 95% 0.025   Confidence Level   100(1-α) α α/2 z α/2 90% 0.10 0.05 1.645 95% 0.025 1.96 99% 0.01 0.005 2.575

Conditions that MUST be met for CI for µ LARGE sample  

Example - In a sample of 100 randomly selected employees, it was found that their mean height was 63.4 inches. From previous studies, it is assumed that the standard deviation, σ, is 2.4. Compute the 95% confidence interval for μ. Check Conditions: Was sample from population a random sample? Yes – stated in problem X-bar normally distributed? Yes – n ≥ 30 (n=100) Confidence Interval for Mean – Large Sample

Example - A group of 49 randomly selected parrots has a mean age of 22 Example - A group of 49 randomly selected parrots has a mean age of 22.4 years with a population standard deviation of 3.8. Compute the 98% confidence interval for μ. Check Conditions: Was sample from population a random sample? Yes – stated in problem X-bar normally distributed? Yes – n ≥ 30 (n=49) Confidence Interval for Mean – Large Sample 0.49 0.49

When σ is not known and n ≥ 30

Example - A random sample of 40 part-time workers has a mean annual earnings of $3120 and a standard deviation of $677. Compute the 95% confidence interval for μ. Check Conditions: Was sample from population a random sample? Yes – stated in problem X-bar normally distributed? Yes – n ≥ 30 (n=40) Confidence Interval for Mean – Large Sample Since we don’t know σ and n ≥ 30

Interpretation of CI Applet: http://bcs.whfreeman.com/ips4e/cat_010/applets/confidenceinterval.html

Interpretation of CI We can be _____% confident that the population mean of ______________________ lies between __________ and ____________.

Example - In a sample of 100 randomly selected employees, it was found that their mean height was 63.4 inches. From previous studies, it is assumed that the standard deviation, σ, is 2.4. Compute the 95% confidence interval for μ. We can be 95% confident that the population mean of heights for employees lies between 62.93 and 63.87 inches.

Example - A group of 49 randomly selected parrots has a mean age of 22 Example - A group of 49 randomly selected parrots has a mean age of 22.4 years with a population standard deviation of 3.8. Compute the 98% confidence interval for μ. We can be 95% confident that the average age of all parrots lies between 21.138 and 23.662 years.

Example - A random sample of 40 part-time workers has a mean annual earnings of $3120 and a standard deviation of $677. Compute the 95% confidence interval for μ. We can be 95% confident that the population mean annual earning of part-time workers lies between $2910.20 and $3329.80.

Confidence interval for µ: Student’s t-statistic Section 5.3 Confidence interval for µ: Student’s t-statistic

With small samples (n < 30) two problems….  

CI for µ SMALL sample

  Based on (n-1) degrees of freedom

  Degrees of freedom = n-1 = 18-1=17

Conditions that MUST be met for CI for µ SMALL sample  

Example - Construct a 99% confidence interval for the population mean, μ. Assume the population has a normal distribution. A group of 19 randomly selected employees has a mean age of 22.4 years with a standard deviation of 3.8 years. Check Conditions: Was sample from population a random sample? Yes – stated in problem X-bar normally distributed? Yes – population is normally distributed Confidence Interval for Mean – Small Sample We are 99% confident that the mean age of all employees lies between 19.891 and 24.909 years.

Example - Construct a 90% confidence interval for the population mean, μ. Assume the population has a normal distribution. A sample of 15 randomly selected math majors has a grade point average of 2.86 with a standard deviation of 0.78. Check Conditions: Was sample from population a random sample? Yes – stated in problem X-bar normally distributed? Yes – population is normally distributed Confidence Interval for Mean – Small Sample We are 90% confident that the mean grade point average of all math majors lies between 2.5053 and 3.2147

Large sample CI for population proportion Section 5.4 Large sample CI for population proportion

RECALL: General Formula for Confidence Interval Point Estimate ± (critical value) (s of distr. of point estimate) [p.e. – c.v(s of p.e), p.e + c.v(s of p.e) ]

   

Large Sample CI for p

Conditions Required  

Example - A university dean is interested in determining the proportion of students who receive some sort of financial aid. Rather than examine the records for all students, the dean randomly selects 200 students and finds that 118 of them are receiving financial aid. Use a 95% confidence interval to estimate the true proportion of students on financial aid.   We are 95% confident that the population proportion of students from the university that receive financial aid is between .5218 and .6582 Confidence Interval for Proportion – Large Sample

Small Sample CI for p  

Example – The International Nanny Association reports that in a random sample of 4,176 nannies who were placed in a job in 2007, only 14 were men. Use Wilson’s adjustment to find a 95% confidence interval for the true proportion of all working nannies who are men. Interpret the resulting interval.   Confidence Interval for Proportion – Small Sample We are 95% confident that the population proportion of working nannies who are men lies between .0019 and .0057.

Should be n+4

Determining the sample size Section 5.5 Determining the sample size

Determining Sample Size Suppose we know Our Confidence Level (ie 95%) An estimate for standard deviation of the population (σ) How close we want our parameter estimate (margin of error) We can determine what size sample we need in order for those criteria to be met…

Margin of Error The value that we want our parameter estimate to be is called the sampling error or margin of error. For CI for µ …

Sample Size Determination for µ Note: value of σ is usually unknown. It can be estimated by the standard deviation from a prior sample (s). OR we can estimate it using σ ≈ range/4.

ALWAYS ROUND UP Sample Size Calculations!

Example - In order to set rates, an insurance company is trying to estimate the average number of sick days that full time workers at a local bank take per year. A previous study indicated that the standard deviation was 2.8 days. a) How large a sample must be selected if the company wants to be 90% confident that the true mean differs from the sample mean by no more than 1 day?

b) Repeat part (a) using a 95% confidence interval b) Repeat part (a) using a 95% confidence interval. Which level of confidence requires a larger sample size? Explain.

Sample Size Determination for p Note: we will not know p but may have a good estimate from a sample OR if no estimate is given use p = .5

 Example - In a college student poll, it is of interest to estimate the proportion p of students in favor of changing from a quarter-system to a semester-system. How many students should be polled so that we can estimate p to within 0.09 using a 99% confidence interval?