Review of Statistical Terms

1. Review of Statistical Terms • Population • Sample • Parameter • Statistic

2. Population the set of all measurements (either existing or conceptual) under consideration

3. Sample a subset of measurements from a population

4. Parameter a numerical descriptive measure of a population

5. Statistic a numerical descriptive measure of a sample

6. We use a statistic to make inferences about a population parameter.

7. Principal types of inferences • Estimate the value of a population parameter • Formulate a decision about the value of a population parameter

8. Sampling Distribution a probability distribution for the sample statistic we are using

9. Example of a Sampling Distribution Select samples with two elements each (in sequence with replacement) from the set {1, 2, 3, 4, 5, 6}.

10. Constructing a Sampling Distribution of the Mean for Samples of Size n = 2 List all samples and compute the mean of each sample. sample: mean: sample: mean {1,1} 1.0 {1,6} 3.5 {1,2} 1.5 {2,1} 1.5 {1,3} 2.0 {2,2} 4 {1,4} 2.5 … ... {1,5} 3.0 There are 36 different samples.

11. Sampling Distribution of the Mean p 1.0 1/36 1.5 2/36 2.0 3/36 2.5 4/36 3.0 5/36 3.5 6/36 4.0 5/36 4.5 4/36 5.0 3/36 5.5 2/36 6.0 1/36

12. Sampling Distribution Histogram | | | | | | | | | | | | 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

13. Let x be a random variable with a normal distribution with mean  and standard deviation . Let be the sample mean corresponding to random samples of size n taken from the distribution .

14. Facts about sampling distribution of the mean:

15. Facts about sampling distribution of the mean: • The distribution is a normal distribution.

16. Facts about sampling distribution of the mean: • The distribution is a normal distribution. • The mean of the distribution is  (the same mean as the original distribution).

17. Facts about sampling distribution of the mean: • The distribution is a normal distribution. • The mean of the distribution is  (the same mean as the original distribution). • The standard deviation of the distribution is  (the standard deviation of the original distribution, divided by the square root of the sample size).

18. We can use this theorem to draw conclusions about means of samples taken from normal distributions. If the original distribution is normal, then the sampling distribution will be normal.

19. The Mean of the Sampling Distribution

20. The mean of the sampling distribution is equal to the mean of the original distribution.

21. The Standard Deviation of the Sampling Distribution

22. The standard deviation of the sampling distribution is equal to the standard deviation of the original distribution divided by the square root of the sample size.

23. The time it takes to drive between cities A and B is normally distributed with a mean of 14 minutes and a standard deviation of 2.2 minutes. • Find the probability that a trip between the cities takes more than 15 minutes. • Find the probability that mean time of nine trips between the cities is more than 15 minutes.

24. Find this area 14 15 Mean = 14 minutes, standard deviation = 2.2 minutes • Find the probability that a trip between the cities takes more than 15 minutes.

25. Mean = 14 minutes, standard deviation = 2.2 minutes • Find the probability that mean time of nine trips between the cities is more than 15 minutes.

26. Find this area 14 15 Mean = 14 minutes, standard deviation = 2.2 minutes • Find the probability that mean time of nine trips between the cities is more than 15 minutes.