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Math 110 Chapter 7 Section 7.1 Inference for the Mean of a Population Inference when is unknown In chapter 6, we made the unrealistic assumption that we knew , the population standard deviation. In practice, is usually unknown and we must use the standard deviation of the sample, s, to estimate . This changes some details of tests and confidence intervals. Since we don't know , we can't calculate the standard deviation of x as . Instead, we n estimate it using the standard error: The standard error of the sample mean x is: s . n When we pretended to know , we based confidence intervals and hypothesis tests for on x - the standardized sample mean z . The z-statistic has the standard normal n distribution. When we substitute standard error of x for its standard deviation, the statistic that results does not have a normal distribution. It has a t-distribution. The t- distribution is bell-shaped, like the normal distribution, but the tails are slightly thicker. There is a different t-distribution for each sample size. We specify a particular t- distribution by giving its degrees of freedom = n - 1. Calculating a t statistic: Draw a random sample of size n from a population that has the normal distribution with mean and standard deviation . The one-sample t statistic x t has the t distribution with n - 1 degrees of freedom. s n Problems: 1. A vending machine dispenses servings of hot chocolate. You buy a sample of 4 servings from the vending machine and measure each serving to obtain the following data (in ml's): 247 246 250 245 . The sample mean x is 247 (verify). a) Find the standard error of the mean: b) What are the degrees of freedom of the t statistic associated with x ? 2. From Table C, find the indicated t* values: a) From the t distribution with 19 degrees of freedom, find the t* having .025 probability to its right. b) From the t distribution with 7 degrees of freedom, find the t* having .90 probability to its left. Math 120 Lecture Notes 62 Confidence Intervals when is unknown A confidence level C confidence interval for an unknown is : s x t* n Exercise: A sample of 5 candies was taken from a bulk food bin. The weights of the candies are: 4.55 4.60 4.41 4.77 4.23 a) Calculate an 80% confidence interval for the population mean weight . b) Calculate a 98% confidence interval for the population mean weight . c) Do you have good evidence that the mean weight of the candies is greater than 4.0 grams? i) State the null and alternate hypothesis ii) Calculate the test statistic, t. iii) Find t* values from table C that bracket the t statistic, hence find an interval that contains the p-value. iv) Decide whether to reject H0 Math 120 Lecture Notes 63 Exercise: Let x = amount of time for a worker in a clothing manufacturing company to join a collar to a shirt. A sample of eight workers gave the following figures for the amount of time (in minutes) needed to join a collar to a shirt: 10 12 13 9 8 14 10 11 Is there enough evidence to conclude that the true mean time to join a collar is more than 10 minutes? (Assume the x's are normally distributed). a) Compute the sample mean x and the sample standard deviation s. State the degrees of freedom: x = __________ s = __________ degrees of freedom = _____ b) State the null and alternate hypotheses c) Compute the test statistic t, and mark it on the horizontal axis of the graph. d) Locate the two critical values t* from Table C that bracket t, and mark them on the graph. e) Between what two values does the p-value of the test fall? Is the t value computed in part c) significant at the = .10 level? At the = .15 level? t distribution with 7 degrees of freedom Math 120 Lecture Notes 64 Exercise: A study on the amount of dye needed to get the best colour for a certain type of fabric was conducted. The following data give the photometer readings for the colour density of the fabric when the amount of dye used was 1% of the weight of the fabric. 13.2 11.5 12.9 13.0 11.7 10.4 12.1 12.1 11.5 10.3 11.7 12.3 11.2 Do the data demonstrate that the true mean photometer reading for the colour density is significantly different from 12? Use 10 percent as the level of significance. (Assume a normal distribution) a) Compute the sample mean x and the sample standard deviation s. State the degrees of freedom: x = __________ s = __________ degrees of freedom = _____ b) State the null and alternate hypotheses c) Compute the test statistic t, and mark it on the horizontal axis of the graph. d) From table C, find the t* value associate with = .10 for a two tailed test. Mark it on the graph. e) Do you have enough evidence to reject H0 at the 10% level of significance? t distribution with 12 degrees of freedom Math 120 Lecture Notes 65 Matched Pairs t Procedure One common experiment design is to compare the response to two treatments given to the same subjects, or to make before and after observations on the same subject. We can determine whether or not there is a significant difference between two treatments by taking differences of the two observations. If there is not any systematic difference in the response to the two treatments, we would expect the mean of the differences to be zero. Example: We would like to know whether most people prefer Brand P (Schweppes) or Brand Q (Western Family) raspberry ginger ale. A Math 120 class was given a sample of each brand and were asked to rate the brands The cups were labeled P and Q. After the students tasted both, they gave each drink a score on scale of 1 to 10. The higher the score, the better they liked the drink. Score for Score for Differences a) Calculate the differences Brand P (x1) Brand Q (x2) (x1 - x2) x1 - x2 7 4 10 4 b) Find the sample mean x of 4 2 the differences 7 8 5 1 ___________________ 10 5 5 3 c) Find the sample standard 4 7 deviations of the differences, 8 10 s 4 1 7 2 ____________________ 6 2 5 5 a) Find a 95% confidence interval for the unknown parameter . b) Do you believe that there is enough evidence to conclude that most people like Brand P better than Brand Q ? (use = 0.05 level of significance) Math 120 Lecture Notes 66 Example: Right-handed subjects were tested for reaction times with their left and right hands. The results (in thousandths of a second) are given below. Subject: A B C D E F G H I J K L M N Right: 191 97 116 165 116 129 171 155 112 102 188 158 121 133 Left: 224 171 191 207 196 165 177 165 140 188 155 219 177 174 a) Use a 0.05 level of significance to test the claim that there is a difference between the mean of the right- and left-hand reaction times. b) Construct a 95% confidence interval to estimate the mean of the differences. Math 120 Lecture Notes 67 Math 120 Section 7.2 Comparing Two Means Often it is useful to compare two means from two different populations. To do this, we take a random sample from each population, and calculate a separate x and s for each sample. The two samples should be independent--that is, one sample has no influence on the other. (this is not the same as matched pairs testing, were we took the differences then found x and s for the differences.) If populations are normally distributed, then the following formulas may be used: Confidence interval for the difference of two means 1 - 2: 2 2 s1 s2 (x1 x2 ) t * (degrees of freedom = smaller of n1 – 1 , n2 – 1) n1 n2 To test the hypothesis H0: 1 = 2 , compute the two-sample t test statistic x1 x 2 t (degrees of freedom = smaller of n1 – 1 , n2 – 1) 2 2 s1 s2 n1 n 2 Example: The heights (measured in inches) of 20 randomly selected women and 30 randomly selected men were independently obtained from the student from the college population. The sample information is given below. Assume that the heights are approximately normally distributed for both populations. Find a 95% confidence interval for the difference between the mean heights: 1 - 2. Male: n1 = 30 x 1 = 69.8 s1 = 1.92 Female: n2 = 20 x 2 = 63.8 s2 = 2.18 Math 120 Lecture Notes 68 Example: Many students have complained that the soft-drink vending machine A dispenses less drink than machine B. To test this belief, a student randomly samples several servings from each machine and carefully measured them, with the results as shown: Machine A: n1 = 10 x 1 = 5.38 s1 = 1.59 Machine B: n2 = 12 x 2 = 5.92 s2 = 0.83 Does this evidence support the hypothesis that the mean amount dispensed by Machine A is less than the amount dispensed by B? Assume the amounts dispensed by both machines are normally distributed and complete the test using = 0.05. Math 120 Lecture Notes 69