MATH 2311 Section 7.4

Confidence Interval for a Population Mean

So, what is t*?

How do we find critical values for a t-distribution? Look at the online text book, under appendices. Degrees of freedom are on the left and the top end (1-confidence level/2) are given at the top.

Using R-Studio to find t* values: The command is qt(1.##/2,df) If you are using a confidence level of 90% with 20 degrees of freedom, you would use the command: qt(1.90/2,20) In the TI 83/84 Calculators, Select 2nd, DISTR, and select invT. invT(1.90/2,20)

Assumptions:

Changing which factors will cause the width of the confidence interval to increase?

Examples:

Popper 22 You select a sample of 50 people with a mean height of 72 inches from a population that has a standard deviation of 3 inches. 1. What would the margin of error be for a 95% confidence interval? 0.698 b. 0.832 c. 16.748 d. 1.644 2. What is the width of confidence interval? 1.396 b. 0.349 c. 1.66 d. 0.415 3. What is the confidence interval? a. [71.17, 72.83] b. [69.43, 75.11] c. [72.41, 75.21] d. [47.31, 53.65]

Popper 22 Continued: 4. Give an interpretation of this interval: A randomly selected person from the population has a 95% chance being within the interval. There is a 95% chance that the population mean will fall within the interval. If numerous samples are selected, 95% of them will contain the population mean. All of these choices 2 of the choices a, b or c. What will cause the width of the confidence interval to increase? decrease n b. decrease confidence level c. decrease standard deviation d. increase the mean

Examples:

t-score table:

t-score table: Since n = 8, we are going to use the value for n – 1 = 7 degrees of freedom. Since we want a confidence level of 95%, we will use an “Upper Tail Probability” of (1 – 0.95)/2 = 0.025. This is to account for only using one tail end of our graph, rather than 2. Our t* value will be 2.3646.

Confidence Interval Calculation:

Look at the following example: The effect of exercise on the amount of lactic acid in the blood was examined in an article for an exercise and sport magazine. Eight males were selected at random from those attending a week-long training camp. Blood lactate levels were measured before and after playing three games of racquetball, as shown in the accompanying table. Use this data to estimate the mean increase in blood lactate level using a 95% confidence interval.

Values to cut and paste: assign(“before”,c(13,20,17,13,13,16,15,16)) assign(“after”,c(18,37,40,35,30,20,33,19))

Interpretation of the Confidence Interval:

Example: A 95% confidence interval for the mean of a population is to be constructed and must be accurate to within 0.3 unit. A preliminary sample standard deviation is 2.9. Find the he smallest sample size n that provides the desired accuracy.

Popper 23: A group of 10 students are comparing exam scores for a test they recently took. Based on these results, you want to find out, with 90% confidence, what range of grades you can expect to receive. Exam Scores: 70, 75, 80, 87, 90, 90, 93, 94, 96, 100 1. What is the sample mean? a. 90 b. 87.5 c. 80 d. 91.6 2. What is the sample standard deviation? a. 9.62 b. 92.5 c. 5.65 d. 13.83 3. What is the t* value that can be used here? a. 1.3722 b. 1.3968 c. 1.8331 d. 1.6449 4. What is the margin of error? a. 5.58 b. 5.00 c. 4.03 d. 6.12 5. What is the confidence interval? a. [70,100] b. [87.5, 100] c. [85.67, 89.33] d. [81.92, 93.08]