PHILIPPINE STATISTCICS UNIT~2.POWER POINT T SAMPLE
- 1. CONFIDENCE INTERVALS FOR THE POPULATION MEAN WHEN IS UNKNOWN
- 2. Lesson Objectives At the end of this lesson, you are expected to: • identify the appropriate distribution when the population σ is unknown; • understand the t-distribution; • state the difference between a z-distribution and a t-distribution; and • identify the confifidence coeffifficients for computing t from the t-Table.
- 4. Lesson Introduction Aldrei wants to know if cooperative grouping is an effective strategy in improving the mathematics performance of Grade 7 students. Twenty students were included in the experimental group while another 20 students were included in the control group. The mean achievement score of the students in the experimental group was 82.5 with a standard deviation of 3 while the mean of the students in the control group was 80 with a standard deviation of 6. The two groups come from normally distributed populations. The confidence level adopted was 95%. • What is the estimate of the population mean where the experimental group comes from? ___________________________ • What is the estimate of the population mean where the control group comes from? ___________________________ • Express your confidence as percentage. ____________________
- 5. Discussion Points Assumptions in Computing for the Population Mean When σ is Unknown When n ≥ 30, and σ is unknown, the sample standard deviation scan be substituted for σ. However, the following assumptions should be met. •The sample is a random sample. •Either n ≥ 30 or the population is normally distributed when n < 30.
- 6. Discussion Points General expression for the confidence interval when σ is unknown The distribution of values is called t-distribution X t s n
- 7. Discussion Points Degrees of Freedom • The degrees of freedom, denoted by df, are the number of values that are free to vary after a sample statistic has been computed • Indicate the specific curve to use when a distribution consists of a family of curves.
- 8. Discussion Points Formula for computing the confidence interval using the t-distribution X t s n X t s n
- 10. Example 1 An admission officer of an educational institution wants to know the mean age of all entering mathematics majors. He computed a mean age of 18 years and a standard deviation of 1.2 years on a random sample of 25 entering mathematics majors purportedly coming from a normally distributed population. With 99% confidence, find the point estimate and the interval estimate of the population mean.
- 14. Exercises 1. Using the t-table, give the confidence coefficients for each of the following: n = 12, 95% confidence n = 15, 95% confidence n = 21, 99% confidence n = 23, 95% confidence n = 25, 99% confidence
- 15. Exercises 2. The mean scores of a random sample of 17 students who took a special test is 83.5. If the standard deviation of the scores is 4.1, and the sample comes from an approximately normal population, find the point and the interval estimates of the population mean adopting a confidence level of 95%. 3. The mean age of 20 youth volunteers in a community project is 17.5 years with a standard deviation of 2 years. If the sample comes from an approximately normal distribution, what are the point and the interval estimates of the population mean? Use 99% confidence level.
- 16. Exercises 4. The average weight of 25 chocolate bars selected from a normally distributed population is 200 g with a standard deviation of 10 g. Find the point and the interval estimates using 95% confidence level.
- 17. Summary General expression for the confidence interval when σ is unknown The distribution of values is called t-distribution X t s n
- 18. Summary