1. When σ is Unknown The One – Sample Interval For a Population Mean Target Goal: I can construct and interpret a CI for a population mean when σ is unknown. I can carry out the 4 step process for confidence intervals. 8.3b h.w: pg. 518: 57, 59, 63 (4 step, show work. Do not say “given in the stem”.)

2. Inference for the Mean of a Population If our data comes from a simple random sample (SRS) and the sample size is sufficiently large, then we know that the sampling distribution of the sample means is approximately normal with mean μ and standard deviation

3. PROBLEM: If σ is unknown, then we cannot calculate the standard deviation for the sampling model.  We must estimate the value of σ in order to use the methods of inference that we have learned.

4. SOLUTION: We will use s (the standard deviation of the sample) to estimate σ. Then the standard error of the sample mean is (referred to as SE or SEM).

5. Recall: when we know σ, we base confidence intervals and tests for μ on the one sample z statistic. has the normal distribution N( 0, 1)

6. PROBLEM: When we do not know σ, we replace for. The statistic that results has more variation and no longer has a normal distribution, so we cannot call it z. It has a new distribution called the t distribution.

7. One-Sample t Statistic has the t distribution with n-1 degrees of freedom. t is a standardized value. Like z, t tells us how many standardized units is from the mean μ.

8. When we describe a t distribution we must identify its degrees of freedom because there is a different t statistic for each sample size. The degrees of freedom (df) for the one- sample t statistic is (n – 1). The t distribution is symmetric about zero and is bell-shaped, but there is more variation so the spread is greater.

9. As the degrees of freedom increase, the t distribution gets closer to the normal distribution, since s gets closer to σ. There is more area in the tails of t distributions. As df increases, the distribution approaches “normal”. t curve for 2 df z curve Why is the z curve taller than the t curve for 2 df?

10. Ex. Using the “t-table” Table B is used to find critical values t* with known probability to its right! What critical value t* would you use for a t dist with 18 df, having a probability 0.90 to the left of t*?.90 corresponds with upper tail probability of.10 so, t* = 1.330 Try: invT(.90, 18)

11. Using Table B to Find Critical t* Values Suppose you want to construct a 95% confidence interval for the mean µ of a Normal population based on an SRS of size n = 12. What critical t* should you use? Estimating a Population Mean In Table B, we consult the row corresponding to df = n – 1 = 11. The desired critical value is t * = 2.201. We move across that row to the entry that is directly above 95% confidence level. Upper-tail probability p df.05.025.02.01 101.8122.2282.3592.764 111.7962.2012.3282.718 121.7822.1792.3032.681 z*1.6451.9602.0542.326 90%95%96%98% Confidence level C

12. t Confidence Intervals and Tests We can construct a confidence interval using the t distribution in the same way we constructed confidence intervals for the z distribution. A level C confidence interval for μ when σ is not known is: Remember, the t Table uses the area to the RIGHT of t*. t* is the upper (1-C)/2 critical value for the t(n-1) distribution

13. Ex. Auto Pollution (C.I. for one sample t-test). Ex. Auto Pollution (C.I. for one sample t-test). Read as class bottom page 509

14. Construct a 95% C.I. for the mean amount of NOX emitted. Step 1: State - Identify the population of interest and the parameter you want to draw a conclusion about. We want to estimate the true mean amount µ of NOX emitted by all light duty engines of this type at a 95% confidence level.

15. Step 2. Choose the appropriate inference procedure. Plan- Since σ is not known, we should construct a one-sample t interval for µ if the conditions are met. Verify the conditions. (Plot data when possible.)

16. Plot data: (statplot,data L1,data axis: X) If the data are normally distributed, the normal probability plot will be roughly linear..

17. Random: The data come from a “random sample” of 40 engines from the population of all light duty engines of this type. Normal: We don’t know whether the population is normal but because the sample size, n = 40, is large (at least 30), the CLT tells us the distribution is approximately normal.

18. Independent: We are sampling without replacement, so we need to check the 10% condition; we must assume that there at least 10(40) = 400 light duty engines of this type.

19. Step 3. Carry out the inference procedure. DO - Given =, df = There is no row for 39,use the more conservative df = 30 which is t* = 2.042 (this gives a higher critical value and wider c.i). The 95% Confidence interval for μ is 1.267540 -1 = 39. = (1.1599, 1.3751)

20. Step 4: Interpret your results in the context of the problem. We are 95% confident that the true mean level of NOX emitted by all light duty engines is between 1.1599 grams/mile and 1.3751 grams/mile. Since the entire interval exceeds 1.0, it appears that this type of engine violates EPA limits.

21. Read pg. 501 - 511