Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Hmmβ¦déjà vu? 0 Statistic is a numerical descriptive measure of a sample 0 Parameter is a numerical descriptive measure of a population Soβ¦letβs do this 0 What are the symbols for Statistic mean, variance, standard deviation? Something new 0 Proportion 0 π (π βππ‘) Note: 0 We are going from raw data distribution to a sampling distribution Note #2: 0 We often do not have access to all the measurements of an entire population because of constraints on time, money, or effort. So we must use measurements from a sample. Type of inferences 0 1) Estimation: In this type of inference, we estimate the value of a population parameter 0 2) Testing: In this type of inference, we formulate a decision about the value of a population parameter 0 3) Regression: In this type of inference, we make predictions or forecasts about the value of a statistical variable Sampling distribution 0 It is a probability distribution of a sample statistic based on all possible simple random samples of the same size from the same population Group Work: 0 1) What is population parameter? Give an example 0 2) What is sample statistic? Give an example 0 3) What is a sampling distribution? Answer 0 1) Population parameter is a numerical descriptive measure of a population 0 2) A sample statistic or statistic is a numerical descriptive measure of a sample 0 3) A sampling distribution is a probability distribution for the sample statistic we are using Read page 295-298 Homework Practice 0 Pg 298-299 #1-9 Central Limit Theorem Central Limit Theorem 0 For a Normal Probability Distribution: 0 Let x be a random variable with a normal distribution whose mean is π and whose standard deviation is π. Let π₯ be the sample mean corresponding to random samples of size n taken from the x distribution. Then the following are true: 0 A) The π₯ distribution is a normal distribution 0 B) The mean of the π₯ distribution is π 0 C) The standard deviation of the π₯ distribution is π/ π Note: 0 We conclude from the previous theorem that when x has a normal distribution, the π₯ distribution will be normal for any sample size n. Furthermore, we can convert the π₯ distribution to the standard normal z distribution by using these formulas: 0 ππ₯ = π 0 ππ₯ = 0 π§= π π π₯βππ₯ ππ₯ = π₯βπ π/ π 0 Where n is the sample size 0 π is the mean of the π₯ distribution, and 0 π is the standard deviation of the x distribution Example 0 Suppose a team of biologist has been studying the height in human. Let x be the height of a single person. The group has determined that x has a normal distribution with mean π = 5.6 ππππ‘ and standard deviation π = 1.5 feet . 0 A) What is the probability that a single person taken at random will be in between 4.7 and 6.5 feet tall? 0 B) What is the probability that the mean length π₯ of 5 people taken at random is between 4.7 and 6.5 feet tall? Answer 0 A) π§ = π₯βπ π = π₯β5.6 1.5 4.7β5.6 1.5 0 π 4.7 < π₯ < 6.5 = π( 0 B) π§ = π₯βππ₯ ππ₯ 4.7β5.6 0 π( 1.5 5 = π₯βπ π₯β5.6 = π/ π 1.5/ 5 <π₯< 6.5β5.6 1.5 5 ) <x< 6.5β5.6 ) 1.5 Group Work 0 Suppose a team of feet analysts has been studying the size of manβs foot for particular area. Let x represent the size of the foot. They have determined that the size of the foot has a normal distribution with π = 8.4 πππβππ and standard deviation π = 3.1 inches. 0 A) Whatβs the probability of the foot of a single person taken at random will be in between 6 and 8 inches? 0 B) Whatβs the probability that the mean size of 8 people taken at random will be in between 7 and 9 inches? Standard error 0 Standard error is the standard deviation of a sampling distribution. For the π₯ sampling distribution, 0 Standard error = ππ₯ = π/ π Using Central limit theorem to convert the π₯ distribution to the standard normal 0π =π distribution π₯ 0 ππ₯ = 0π§= π π π₯βππ₯ ππ₯ = π₯βπ π/ π 0 Where n is the sample size (π β₯ 30), 0 π is the mean of the x distribution, and 0 π is the standard deviation of the x distribution Group Work: Central Limit Theorem 0 A) Suppose x has a normal distribution with population mean 18 and standard deviation 3. If you draw random samples of size 5 from the x distribution and x bar represents the sample mean, what can you say about the x bar distribution? How could you standardize the x bar distribution? 0 B) Suppose you know that the x distribution has population mean 75 and standard deviation 12 but you have no info as to whether or not the x distribution is normal. If you draw samples of size 30 from the x distribution and x bar represents sample mean, what can you say about the x bar distribution? How could you standardize the x bar distribution? 0 C) Suppose you didnβt know that x had a normal distribution. Would you be justified in saying that the x bar distribution is approximately normal if the sample size were n=8? Answer 0 A) Since you are given it to be normal, the x bar distribution also will be normal even though sample π₯β18 size is much less than 30. π§ = 3/ 5 0 B) Since sample size is large enough, the x bar distribution will be an approximately normal π₯β75 distribution. π§ = 12/ 30 0 C) No, sample size is too small. Need to be 30 or more Note: 0 A sample statistic is unbiased if the mean of its sampling distribution equals the values of the parameter being estimated 0 The spread of the sampling distribution indicates the variability of the statistic. The spread is affected by the sampling method and the sample size. Statistics from larger random samples have spread that are smaller. Read 304 and 305 Homework Practice 0 Pg 306-308 #1-18 eoe Sampling Distributions for Proportions Think Back to Section 6.4 0 We dealt with normal approximation to the binomial. 0 How is this related to sampling distribution for proportions? 0 Well in many important situations, we prefer to work with the proportion of successes r/n rather than the actual number of successes r in binomial experiments. Sampling distribution for the π proportion π = 0 0 0 0 0 Given n= number of binomial trials (fixed constant) r= number of successes p=probability of success on each trial q=1-p= probability of failure on each trial 0 If np>5 and nq>5, then the random variable π = π π π can be approximated by a normal random variable (x) with mean and standard deviation 0 ππ = π πππ ππ = ππ π Standard Error 0 The standard error for the π distribution is the standard deviation ππ = ππ π Where do these formula comes from? 0 Remember π is as known as expected value or average. 0π= π₯ π(π₯) 0 So ππ = 0 ππ = ππ π ππ π = ππ π = πππ π = π (r is number of success) = ππ π How to Make continuity corrections to π intervals 0 If r/n is the right endpoint of a π interval, we add 0.5/n to get the corresponding right endpoint of the x interval 0 If r/n is the left endpoint of a π interval, we subtract 0.5/n to get the corresponding left endpoint of the x interval Example: 0 Suppose n=30 and we have a π interval from 15/30=0.5 and 25/30=.83 Use the continuity correction to convert this interval to an x interval Answer 0 0.5/30 = 0.02 (approx) 0 So x interval is .5-.02 and .83+.02 which is .48 to .85 0 π interval: .5 to .83 0 x interval: .48 to .85 Group Work 0 Suppose n=50 and π interval is .64 and 1.58. Find the x interval Word Problem 0 Annual cancer rate in L.A. is 209 per 1000 people. Suppose we take 40 random people. 0 A) What is the probability p that someone will get cancer and whatβs the probability q that they wonβt get cancer? 0 B) Do you think we can approximate π with a normal distribution? Explain 0 C) What are the mean and standard deviation for π? 0 D) What is the probability that between 10% and 20% of the people will be cancer victim? Interpret the result Answer 0 A) π = 209 1000 = .209 πππ π = .791 0 B) ππ = 40 .209 = 8.36 πππ ππ = 40 .791 = 31.64 0 Since both are greater than 5, we can approximate π with a normal distribution 0 C) ππ = π = .209 0 ππ = ππ π = .209 .791 40 = 0.064 0 D) Since probability is .10 β€ π β€ .20 we need to convert into x distribution 0 Continuity correction=0.5/n=0.5/40=0.0125, so we subtract .01 and add .01 from the interval 0 So .09 β€ π₯ β€ .21, then convert this into z value. 0 π 0.09β.209 .064 β€π§β€ .21β.209 .064 , π‘βππ π’π π π‘ππππ π‘π ππππ π‘βπ ππππππππππ‘π¦ Group work 0 In the OC, the general ethnic profile is about 47% minority and 53% Caucasian. Suppose a company recently hired 78 people. However, if 25% of the new employees are minorities then there is a problem. What is the probability that at most 25% of the new fires will be minorities if the selection process is unbiased and reflect the ethnic profile? (Follow the last exampleβs footstep) Answer 0 π = 78, π = .47, π = .53 0 π = .25 0 0 ππ = 78 .47 = 36.66 πππ ππ = 78 .53 = 41.34 Since both are greater than 5, so normal approximation is appropriate 0 ππ = π = .47 0 ππ = 0 Continuity correction 0.5/n = 0.5/78 = .006 0 Since it is to the right endpoint, you add .006 0 P(π β€ .25) = π(π₯ β€ .256) 0 π π§β€ ππ π = .256β.47 .06 .47 .53 78 = β .06 Remember Control Chart? 0 Sketch how a control chart looks like P-Chart 0 It is just like the control chart How to make a P-Chart? 0 1. Estimate π, π‘βπ ππ£πππππ ππππππππ‘πππ ππ π π’ππππ π ππ 0 π= π‘ππ‘ππ ππ’ππππ ππ πππ πππ£ππ π π’ππππ π ππ ππ πππ π ππππππ πππ‘ππ ππ’ππππ ππ π‘πππππ ππ πππ π ππππππ 0 2. The center like is assigned to be ππ = π 0 3. Control limits are located at π ± 2 ππ/π (2 ππ‘ππππππ π·ππ£πππ‘ππππ ) and π ± 3 ππ/π(3 ππ‘ππππππ π·ππ£πππ‘ππππ ) 0 4. Interpretation: out-of-control signals 0 A) any point beyond a 3 standard deviation level 0 B) 9 consecutive points on one side of the center line 0 C) at least 2 out of 3 consecutive points beyond a 2 standard deviation level 0 Everything is in control if no out-of-control signals occurs Example situation: 0 Civics and Economics is taught in each semester. The course is required to graduate from high school, so it always fills up to its maximum of 60 students. The principal asked the class to provide the control chart for the proportion of Aβs given in the course each semester for the past 14 semesters. Make the chart and interpret the result. Example: Semester 1 2 3 4 5 6 7 r=#of Aβs 9 12 8 15 6 7 13 π=r/60 .15 .20 .13 .25 .1 .12 .22 Semester 8 9 10 11 12 13 14 r=#of Aβs 7 11 9 8 21 11 10 π=r/60 .12 .18 .15 .13 .35 .18 .17 Answer 0 π= π‘ππ‘ππ ππ’ππππ ππ πππ πππ£ππ π π’ππππ π ππ ππ πππ π ππππππ πππ‘ππ ππ’ππππ ππ π‘πππππ ππ πππ π ππππππ (this is pooled proportion of success) 0 n=60 0 π= 147 840 0 ππ = = .175, π = .825 ππ π = .175β.825 60 = .049 0 Control Limits are: .077 and .273 (2 standard deviation) 0 .028 and .322 (3 standard deviation) 0 Then graph it! Y-axis is proportion, and x-axis is sample number Group Work 0 Mr. Liu went on a streak of asking 30 women out a month for 11 months. Complete the table and make a control chart for Mr. Liu and interpret his βgameβ. month 1 2 3 4 5 6 7 1 3 15 12 20 5 8 9 10 11 r=says yes 2 8 7 23 r=says yes 10 π=r/30 month π=r/30 Homework Practice 0 P317-320 #1-13 eoo