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AP Statistics Thursday, 13 February 2014

AP Statistics Thursday, 13 February 2014. OBJECTIVE TSW explore Hypothesis Testing. TEST: Confidence Intervals is not graded. Hypothesis Tests. One Sample Means. Take a sample & find x.

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AP Statistics Thursday, 13 February 2014

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  1. AP StatisticsThursday, 13 February 2014 • OBJECTIVETSW explore Hypothesis Testing. • TEST: Confidence Intervals is not graded.

  2. Hypothesis Tests One Sample Means

  3. Take a sample & find x. But how do I know if this x is one that I expect to happen or is it one that is unlikelyto happen? How can I tell if they really are underweight? Example:A government agency has received numerous complaints that a particular restaurant has been selling underweight hamburgers. The restaurant advertises that it’s patties are “a quarter pound” (4 ounces). Hypothesis testing will help me decide!

  4. What are hypothesis tests? Calculations that tell us if a value occurs by random chance or not. If it is statistically significant, is it . . . • a random occurrence due to variation? • a biased occurrence due to some other reason?

  5. Nature of hypothesis tests - How does a murder trial work? • First begin by supposing the “effect” is NOT present • Next, see if data provides evidence against the supposition Example: murder trial First - assume that the person is innocent Then – must have sufficient evidence to prove guilty Hmmmmm … Hypothesis tests use the same process!

  6. Notice the steps are the same except we add hypothesis statements – which you will learn today Steps: • Assumptions • Hypothesis statements & define parameters • Calculations • Conclusion, in context

  7. Assumptions for z-test (t-test): Have an SRSof context Distribution is (approximately) normal Given Large sample size Graph data s is known (unknown) YEA – These are the same assumptions as confidence intervals!!

  8. Have an SRS of bottles • Sampling distribution is approximately • normalbecause the boxplot is • symmetrical • s is unknown Example 1:Bottles of a popular cola are supposed to contain 300 mL of cola. There is some variation from bottle to bottle. An inspector, who suspects that the bottler is under-filling, measures the contents of six randomly selected bottles. Are the assumptions met? 299.4 297.7 298.9 300.2 297 301

  9. Writing Hypothesis statements: • Null hypothesis – is the statement being tested; this is a statement of “no effect” or “no difference” • Alternative hypothesis – is the statement that we suspect is true H0: Ha:

  10. The form: Null hypothesis H0: parameter = hypothesized value Alternative hypothesis Ha: parameter > hypothesized value Ha: parameter < hypothesized value Ha: parameter ≠hypothesized value or or

  11. Example 2: A government agency has received numerous complaints that a particular restaurant has been selling underweight hamburgers. The restaurant advertises that it’s patties are “a quarter pound” (4 ounces). State the hypotheses : You MUST indicate what μ represents! H0: m = 4 Ha: m < 4 Where m is the true mean weight of hamburger patties

  12. Example 3: A car dealer advertises that his new subcompact models get 47 mpg. You suspect the mileage might be overrated. State the hypotheses : H0: m = 47 Ha: m < 47 Where m is the true mean mpg

  13. Example 4: Many older homes have electrical systems that use fuses rather than circuit breakers. A manufacturer of 40-A fuses wants to make sure that the mean amperage at which its fuses burn out is in fact 40. If the mean amperage is lower than 40, customers will complain because the fuses require replacement too often. If the amperage is higher than 40, the manufacturer might be liable for damage to an electrical system due to fuse malfunction. State the hypotheses : H0: m = 40 Ha: m ≠ 40 Where m is the true mean amperage of the fuses

  14. Facts to remember about hypotheses: • ALWAYS refer to populations (parameters) • The null hypothesis for the “difference” between populations is usually equal to zero • The null hypothesis for the correlation (rho) of two events is usually equal to zero. H0: mx-y= 0 H0: r= 0

  15. Must use parameter (population) x is a statistic (sample) Activity: For each pair of hypotheses, indicate which are not legitimate & explain why: Must be NOT equal! p is the population proportion! Must use same number as H0! r is parameter for population correlation coefficient – but H0MUST be “=“ !

  16. Assignment • WS Hypothesis Testing #1 • Due in 30 minutes.

  17. JVHS Falcons

  18. AP StatisticsFriday, 14 February 2014 • OBJECTIVETSW explore Hypothesis Testing.

  19. Hypothesis Testing WS #1 1a) No, H0 must be = b) No, must use parameter μ c) Yes 2a) H0: μ = 170 Where μ is the true mean cholesterol level in Japanese Ha: μ < 170 children b) H0: μ = 30 ppm Where μ is the true mean nitrate concentration in the Ha: μ > 30 ppm water c) H0: μ = 2.6 hrs Where μ is the true mean response time of service Ha: μ≠2.6 hrstechnicians d) H0: μ = $42,500 Where μ is the true mean household income of mall Ha: μ > $42,500 shoppers e) H0: μ = 5 mm Where μ is the true mean diameter of the spindles Ha: μ≠5 mm 3a) Yes, the normal probability (quantile) plot is approximately linear so the distribution is approximately normal. b) Yes, the sample size is greater than 30, so normality can be assumed.

  20. Hypothesis Tests One Sample Means

  21. P-values - • The probability that the test statistic would have a value as extreme or morethan what is actually observed In other words . . . is it far out in the tails of the distribution?

  22. Level of significance - • Is the amount of evidence necessary before we begin to doubt that the null hypothesis is true • Is the probability that we will reject the null hypothesis, assuming that it is true • Denoted by a • Can be any value • Usual values: 0.1, 0.05, 0.01 • Most common is 0.05

  23. Statistically significant – • The p-value is as small or smaller than the level of significance (a) • If p > a, “fail to reject” the null hypothesis at the a level. • If p <a, “reject” the null hypothesis at the a level.

  24. Facts about p-values: • ALWAYSmake a decision about the null hypothesis! • Large p-values show support for the null hypothesis, but never that it is true! • Small p-values show support that the null is not true. • Double the p-value for two-tail (≠)tests • Never accept the null hypothesis!

  25. Never“accept” the null hypothesis! Never“accept” the null hypothesis! Never“accept” the null hypothesis!

  26. At an alevel of 0.05, would you reject or fail to reject H0 for the given p-values? • 0.03 • 0.15 • 0.45 • 0.023 Reject Fail to reject Fail to reject Reject

  27. Calculating p-values • For z-test statistic – • Use normalcdf(lb,ub) • [using standard normal curve] • For t-test statistic – • Use tcdf(lb, ub, df)

  28. Draw & shade a curve & calculate the p-value: 1) right-tail test t = 1.6; n = 20 2) left-tail test z = -2.4; n = 15 3) two-tail test t = 2.3; n = 25 p = 0.06305 p = 0.008198 p = 0.03045

  29. Writing Conclusions: • A statement of the decision being made (reject or fail to reject H0) & why (linkage) • A statement of the results in context. (state in terms of Ha) AND

  30. “Since the p-value < (>) a, I reject (fail to reject) the H0. There is (is not) sufficient evidence to suggest that Ha.” Be sure to write Ha in context (words)!

  31. P-value = tcdf(2.1,10^99,24) =0.0232 t=2.1 H0: m = 15 Ha: m > 15 Where m is the true mean concentration of lead in drinking water Example 5: Drinking water is considered unsafe if the mean concentration of lead is 15 ppb (parts per billion) or greater. Suppose a community randomly selects of 25 water samples and computes a t-test statistic of 2.1. Assume that lead concentrations are normally distributed. Write the hypotheses, calculate the p-value & write the appropriate conclusion for a = 0.05. Since the p-value < a, I reject H0. There is sufficient evidence to suggest that the mean concentration of lead in drinking water is greater than 15 ppb.

  32. P-value = tcdf(1.9,10^99,11) =0.0420 t=1.9 H0: m = 240 calories Ha: m > 240 calories Where m is the true mean caloric content of the frozen dinners Example 6: A certain type of frozen dinners states that the dinner contains 240 calories. A random sample of 12 of these frozen dinners was selected from production to see if the caloric content was greater than stated on the box. The t-test statistic was calculated to be 1.9.(Assume calories vary normally.) Write the hypotheses, calculate the p-value & write the appropriate conclusion for a = 0.05. Since the p-value < a, I reject H0. There is sufficient evidence to suggest that the true mean caloric content of these frozen dinners is greater than 240 calories.

  33. ASSUMPTIONS • SRS (given) • Normal distribution (given) •  unknown • Ho:  = 240 caloriesHa:  > 240 calories where  is the true mean caloric content of frozen dinners p-value = tcdf(1.9, ∞, 11) = 0.04197 <  = 0.05 Since p ≤ , we reject H0. There is evidence to suggest that the true mean caloric content of frozen dinners is greater than 240 calories.

  34. WS Hypothesis Testing #2 Due, Monday, 17 February 2014.

  35. JVHS Falcons

  36. APStatisticsMonday, 17 February 2014 • OBJECTIVETSW explore the aspects of hypothesis testing. • ASSIGNMENT DUE • WS Hypothesis Testing #2 wire basket • ASSIGNMENT DUE TOMORROW • WS Hypothesis Testing #3 • LOOKING AHEAD • Tuesday, 02/18/2014: Matched Pairs • Thursday, 02/20/2014:QUIZ: Hypothesis Testing REVIEW: Hypothesis Testing • Friday, 02/21/2014: ASSESSMENT: Hypothesis Testing • Monday, 02/24/2014: TEST: Hypothesis Testing

  37. Hypothesis Tests One Sample Means

  38. Formulas: s known: m z =

  39. Formulas: s unknown: m t =

  40. Example 7: The Fritzi Cheese Company buys milk from several suppliers as the essential raw material for its cheese. Fritzi suspects that some producers are adding water to their milk to increase their profits. Excess water can be detected by determining the freezing point of milk. The freezing temperature of natural milk varies normally, with a mean of -0.545 degrees and a standard deviation of 0.008. Added water raises the freezing temperature toward 0 degrees, the freezing point of water (in Celsius). The laboratory manager measures the freezing temperature of five randomly selected lots of milk from one producer with a mean of -0.538 degrees. Is there sufficient evidence to suggest that this producer is adding water to his milk? (Include assumptions.)

  41. SRS? Assumptions: Normal? How do you know? • I have an SRS of milk from one producer • The freezing temperature of milk is a normal distribution. (given) Do you know ? • is known What are your hypothesis statements? Is there a key word? H0: μ= -0.545 Ha: μ> -0.545 where μis the true mean freezing temperature of milk Plug values into formula. p-value = normalcdf(1.9566, 1E99) = 0.0252 Use normalcdf to calculate p-value. α= .05

  42. Compare your p-value to α& make decision Conclusion: Since p-value < α, I reject the null hypothesis. There is sufficient evidence to suggest that the true mean freezing temperature is greater than -0.545. This suggests that the producer is adding water to the milk. Write conclusion in context in terms of Ha.

  43. Example 8: The Degree of Reading Power (DRP) is a test of the reading ability of children. Here are DRP scores for a random sample of 44 third-grade students in a suburban district: (data on note page) At the a = 0.1, is there sufficient evidence to suggest that this district’s third graders reading ability is different than the national mean of 34?

  44. SRS? • I have an SRS of third-graders Normal? How do you know? • Since the sample size is large, the sampling distribution is approximately normally distributed Do you know ? • is unknown What are your hypothesis statements? Is there a key word? H0: μ= 34 where μis the true mean reading Ha: μ≠34 ability of the district’s third-graders Plug values into formula. p-value = 2*tcdf(0.6467, 1E99, 43) = 2*0.2606 = 0.5212 Use tcdf to calculate p-value. α= 0.1

  45. Compare your p-value to α& make decision Conclusion: Since p-value > α, I fail to reject the null hypothesis. There is not sufficient evidence to suggest that the true mean reading ability of the district’s third-graders is different than the national mean of 34. Write conclusion in context in terms of Ha.

  46. Example 9: The Wall Street Journal (January 27, 1994) reported that based on sales in a chain of Midwestern grocery stores, President’s Choice Chocolate Chip Cookies were selling at a mean rate of $1323 per week. Suppose a random sample of 30 weeks in 1995 in the same stores showed that the cookies were selling at the average rate of $1208 with standard deviation of $275. Does this indicate that the sales of the cookies is different from the earlier figure? (Include assumptions.)

  47. Assumptions • Have an SRS of weeks • Distribution of sales is approximately normal due to large sample size • s unknown • H0: μ= 1323 where μis the true mean cookie sales per week • Ha: μ≠ 1323 • Since p-value < αof 0.05, I reject the null hypothesis. There is sufficient evidence to suggest that the sales of cookies are different from the earlier figure.

  48. Example 9 (Continued): President’s Choice Chocolate Chip Cookies were selling at a mean rate of $1323 per week. Suppose a random sample of 30 weeks in 1995 in the same stores showed that the cookies were selling at the average rate of $1208 with standard deviation of $275. Compute a 95% confidence interval for the mean weekly sales rate. (Just compute the interval.) CI = ($1105.30, $1310.70) Based on this interval, is the mean weekly sales rate statistically different from the reported $1323? The sales rate is statistically different, since the reported mean of $1323 is not in the interval.

  49. Assignment • WS Hypothesis Testing #3 • Due on tomorrow, Tuesday, 18 February 2014. • WS Hypothesis Testing #4 • Due on Thursday, 20 February 2014.

  50. JVHS Falcons

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