Statistics Spring Study Guide

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1 Name: Statistics Spring Study Guide NORMAL AND SAMPLING DISTRIBUTIONS CHAPTER SIX As opposed to the discrete distributions in the previous chapter, the normal distribution is a continuous distribution, and its graph is a curve. Real world data centered around a mean, such as weights of people, normally follow normal distributions. By converting raw scores into standardized (z) scores representing the number of standard deviations a raw score is above the mean, probabilities and other percentages can be calculated regarding where a score fits in a distribution. For example, a score of 12 in a normal distribution with µ = 10 and ß = 2 would have a z score of 1 and would be higher than 84% of other scores in the distribution. This value of 84% is the same for z = 1 in any normal distribution, not just µ = 10 and ß = 2. Even if a distribution is not normal, sample means from the distribution do tend to be normally distributed. 6-A The Normal Curve normal curve ➊ Sketch and label a normal curve for a given µ and ß and use it to calculate probabilities or frequencies. ➊ Given the mean height of college men is 69 with standard deviation 3, find the following. a) the percentage of college men between 66 and 75 tall b) the probability that a random one will be shorter than B Normal Probabilities standardized score z chart ➊ Find the area under the normal curve between 0 and z. ➊ Find the area under the normal curve between 0 and z = ➋ Calculate normal probabilities or frequencies from z scores. ➋ Find the probability of a z score falling in the given ranges. a) P(-1.00 < z < 1.63) b) P(1.00 < z < 1.63) c) P(z < 1.63) ➌ Convert a raw score x to a standardized score z. ➌ Calculate the z score a 25-year-old man weighing 75 kg, given µ = 77 kg and ß = 13 kg. ➍ Calculate normal probabilities or frequencies from raw scores. ➍ On a test with normally distributed scores with µ = 140 and ß = 55, what percent of scores are between 195 and 230? 6-C Percentiles and the Normal Curve ➊ Find the z score needed to achieve a given area between 0 and z. ➊ 40% of the area under the normal curve lies between 0 and which z scores?

2 ➋ Convert a standardized score z to a raw score x. ➋ Find the height of a college woman with a z score of given µ = 65 and ß = 2.5. ➌ Calculate the raw score needed to be higher or lower than a specified percentage of the population. ➌ How high of a score is needed on an SAT I to be in the 90 th percentile, given µ = 500 and ß = 100? 6-D Sampling Distributions standard error sampling distribution central limit theorem ➊ Calculate sampling probabilities or frequencies of sample means or sample means from percentiles. ➊ On a test with normally distributed scores with µ = 140 and ß = 55, what percent of 4-score averages are between 195 and 230? CONFIDENCE INTERVALS CHAPTER EIGHT A c% confidence interval for a mean is a range centered around the sample mean that is c% likely to include the population mean. If ß is known, the confidence interval can be found by adding and subtracting z ß / n from the sample mean, where z is the number of standard deviations needed to achieve the desired percentage as in chapter 6. If ß is not known, Student s t distribution is used instead of the normal distribution. This is similar to the normal distribution but more spread out and dependent on sample size. Confidence intervals for proportions can also be calculated. 8-A Confidence Intervals for a Mean confidence interval confidence level critical value t distribution t chart degrees of freedom margin of error ➊ Find a critical value z 0 or t 0. ➊ Find the critical value for the following confidence intervals. a) c = 95%, ß = 8.1, n = 45 b) c = 80%, ß = 0.72, n = 11 c) c = 95%, s = 21, n = 6 ➋ Find a confidence interval for a mean and interpret it in words. ➋ The heights of 6 random botkin bushes after one year average 84 cm with standard deviation 10 cm. Give a 95% confidence interval for population mean height, and interpret the interval in words. 8-B Confidence Intervals for a Proportion ➊ Find a confidence interval for a proportion and interpret it in words. ➊ In a survey of 1500 Americans, 822 say they approve of President Obama. Make a 95% confidence interval for the percentage of all Americans who approve of Obama, and interpret the interval in words.

3 ➋ Interpret a poll result based on its margin of error. ➋ According to a poll by Rasumussen Reports ending February 9, 2013, 29% of likely voters believe the country is on the right track. The poll had 3500 people and a margin of error of ± 2 percentage points. 8-C Sample Size Needed for a Specified Margin of Error ➊ Estimate the sample size needed to achieve a specified margin of error for a confidence interval for a mean. ➊ Joe wants to find the average speed of drivers of drivers on Navarra Drive. Given s = 4.5 mph from a previous survey, how many cars should he record to have a margin of error of 1.5 mph for a 98% confidence interval? ➋ Estimate the sample size needed to achieve a specified margin of error for a confidence interval for a proportion. ➋ Andie wants to find out what percentage of Santa Cruz adults were born here. How many people should she survey to have an 8% margin of error for a 95% confidence interval? HYPOTHESIS TESTING WITH Z AND T CHAPTER NINE Roughly speaking, if a confidence interval is built around a given µ and a sample is collected, there is a high probability (equal to the confidence level) that the sample mean will fall within the confidence interval. If the sample mean falls outside the confidence interval, it could be a coincidence or it could be that the population mean is not the value originally assumed. This general procedure, called hypothesis testing, can be applied in many testing situations, including comparing two populations to see if there is a significant difference between their parameters. It provides the foundation of social science research, but it is not without its drawbacks. In particular, since it is based on probability, any conclusion always has a possibility of being incorrect; thus nothing can ever be proved with statistics. 9-A Introduction to Hypothesis Testing inferential statistics hypothesis testing null hypothesis alternate hypothesis left-tailed test right-tailed test two-tailed test accept the null reject the null statistically significant type I error type II error power ➊ State the null and alternate hypothesis for a test of a single mean or proportion. ➊ The average birthweight of Castro cattle is known to be 29 kg. Cameron is testing whether feeding the mother a protein-rich diet before birth causes increased birth weights. ➋ Identify factors leading to uncertainties about conclusions from statistical inference. ➋ Cameron finds the weights (in kg) of six newborn cattle with mothers fed protein-rich diets: 32.5, 30.1, 28.1, 27.4, 29.9, The average weight is more than 29 kg. Why should he be cautious in concluding that protein-rich diets increase birthweight? ➌ State the meaning of a type I error and a type II error for a test. ➌ State possible statistical conclusion errors for Cameron s cows.

4 9-B P Values p value level of significance alpha ➊ Calculate z or t for a sample mean or proportion. ➊ The average weight of a Curtiss cat is 4.13 kg. In Drew s sample of 20 Curtiss cats, = 3.81 kg and s = 0.52 kg. ➊ 65% of Curtiss cats are black. In Drew s sample, 14 were black. ➋ Calculate the p value of a z test of a single mean or proportion, and interpret its meaning. ➋ Big Macs are supposed to have 640 calories with ß = 29. McDonald s Corp. audits Bryce s franchise to make sure Big Macs there are in fact 640 calories on average. In their sample of 34 Big Macs, = 630. ➋ Melinda wants to find out if people have a preference between Big Macs and Whoppers. (That is, she wants to find out if the proportion of people who prefer Big Macs over Whoppers is more than 1 / 2.) In a survey of 80 people, 47 say they prefer Big Macs. ➌ Use a p value to make a statistical conclusion about a test. ➌ Make a conclusion about Bryce s Big Macs. 9-C Critical Values critical value critical region ➊ Find a critical value for a z test. ➊ a) å =.05, left-tailed b) å =.10, two-tailed ➋ Find a critical value for a t test. ➋ n = 20, å =.05, two-tailed ➌ Use a critical value to make a statistical conclusion about a test. ➌ Make a conclusion about Bryce s Big Macs.

5 9-D Results of Statistical Tests ➊ Report the results of a study using a p value range. ➊ Seryna tests mg Advil tablets to verify that µ does in fact equal 200. She finds = with s = E Within-Participants Designs within-participants design ➊ Do a statistical test for a within-participants design. ➊ Justine is testing to see if people can balance better with their eyes open than closed. She has seven participants each see how long they can balance on a special balance board with eyes open and also with eyes closed. Participant #: Eyes open (seconds): Eyes closed (seconds): F Between-Participants Designs between-participants design order effects counterbalancingç ➊ Use a Test function on the calculator to do a statistical test of two means using a between-participants design. ➊ Justine is testing to see if people can balance better with their eyes open. Use the same data as in 9-E. ➋ Determine whether or not a within-participants design is appropriate for a test of two means. ➋ Is the balance study better off done with a within-participants design or a between-participants design? ➌ Use a Test function on the calculator to do a statistical test of two proportions for a between-participants design. ➌ Mueller & Dweck (1998) had fifth-graders do puzzles, and they told some of them that they were smart. Then they gave all of them more difficult puzzles and told them that they did not do well. When asked later about how well they had done, 11 of the 30 who had been told they were smart and 8 of the 58 of the others lied about their score.

6 HYPOTHESIS TESTING WITH R, ç 2, AND F CHAPTER ELEVEN Different types of hypotheses require different types of statistical tests. In addition to z and t which are used for means and proportions, three other common statistics used for statistical tests are r, ç 2, and F. This chapter covers six new types of tests using these statistics. They can be found by hand or by calculator, and this chapter covers both methods. Calculating the statistics by hand has no practical use, as they can much more easily be found by calculator; rather, the purpose of knowing how the values are calculated is to gain an understanding of what the statistics actually represent. The chapter ends with a flowchart to help determine which statistical test to use in various situations. 11-A Relationships Between Two Continuous Variables: Linear Correlation positive correlation negative correlation correlation coefficient rho test of a correlation line of best fit interpolation extrapolation ➊ Estimate r for a correlation. ➊ x = temperature, y = heating bill ➋ Calculate r by hand. ➌ Do an r test of a correlation. ➌ Is there a correlation between average monthly temperature and Joe s monthly heating bill? Month: October November December January February March Average temperature ( F): Heating bill ($): ➍ Find the equation of a line of best fit. ➍ Find the equation of the line of best fit for Joe s data. ➎ Use a line of best fit to predict a value of the dependent variable, and consider the accuracy of the estimate. ➎ Predict Joe s heating bill for a month averaging 84, and discuss the accuracy of the prediction. 11-B Relationships Between Two Discrete Variables: Independence chi-square statistic test of independence cell observed value expected value ➊ Find a critical value of ç 2 for a test of independence. ➊ Darby is testing to see if there is a relationship between type of cell phone and gender. She asks 80 boys and 100 girls whether they use an iphone, an Android, a keypad phone, or other. ➋ Sketch a ç 2 distribution. ➋ Sketch a ç 2 distribution for grade level (freshman, sophomore, junior, senior) versus dances attended this year (none, one, some).

7 ➌ Calculate ç 2 by hand for a test of independence. ➍ Do a ç 2 test of independence. ➍ 15 boys had iphones, 20 had Androids, 18 had keyboard phones, and 27 had other. 34 girls had iphones, 28 had Androids, 20 had keyboard phones, and 18 had other. 11-C Distributions: Goodness of Fit goodness of fit test ➊ Calculate ç 2 by hand for a goodness of fit test. ➊ The distribution of highest education level of Californians over 25 is 19.6% with no diploma, 53.3% with a high school diploma, 22.9% with a bachelor s degree, and 4.2% with a graduate degree. In a random sample of 180 Santa Cruz County residents over 25, 26 have no diploma, 81 have a high school diploma only, 53 have a bachelor s degree only, and 11 have a graduate degree. ➋ Do a ç 2 goodness of fit test. ➋ Is the distribution of education level in Santa Cruz County different from that of California overall? ➌ Test to see if a sample does not come from a normal distribution. ➌ Fall Statistics semester grades last year were 66.0, 68.6, 74.8, 75.0, 75.2, 75.2, 76.4, 76.5, 76.9, 77.7, 78.2, 78.9, 79.7, 80.0, 80.5, 81.4, 82.1, 83.2, 83.4, 83.7, 84.6, 84.6, 84.8, 85.6, 85.9, 86.5, 86.5, 86.7, 86.7, 88.4, 88.9, 89.0, 89.0, 89.6, 90.0, 91.2, 91.3, 91.4, 93.0, 93.1, 93.7, 94.3, 94.4, 94.4, 94.8, 95.7, 95.8, 96.0, 96.1, 96.5, 97.3, 98.0, 98.2, 98.3, 100.2, D Standard Deviations: Single Variance test of a single variance ➊ Calculate ç 2 for a test of a single variance. ➊ Cameron times the life of 25 AA batteries in a motor and finds = 445 minutes and s = 42 minutes. Calculate ç 2 for the difference the standard deviation is from one hour. ➋ Find critical values of ç 2. ➋ Find the critical values for Cameron s test in ➊ (df = 24) for the tails indicated. a) right b) left c) two

8 ➌ Do a ç 2 test of a single variance. ➌ Can Cameron conclude that the standard deviation of AA batteries is different from 60 minutes? 11-E Differences Between Standard Deviations: Two Variances F statistic test of two variances ➊ Calculate F for a test of two variances. ➊ Noah has two different routes he can take to school. He times Route A 10 times and finds a standard deviation of s = 99 seconds. He times Route B 13 times and finds a standard deviation of s = 132 seconds. ➋ Find a critical value of F. ➋ Find the critical value for a two-tailed test of Noah s data in ➊. ➌ Do an F test of two variances. ➌ Can Noah conclude that one route has a smaller standard deviation than the other? 11-F Differences Between Means: ANOVA analysis of variance (ANOVA) means square ➊ Calculate the sum of squares within samples. ➋ Calculate the sum of squares between samples. ➌ Calculate the variance between samples. ➍ Calculate the variance within samples. ➎ Calculate F for an ANOVA. ➏ Do an ANOVA. ➏ Karissa plants eight trees throughout Santa Cruz, seven throughout San Jose, and five throughout San Diego. After one year their heights (in cm) are: SC 43, 48, 51, 47, 41, 44, 52, 50; SJ 45, 51, 55, 50, 48, 49, 45; SD 49, 56, 55, 53, G Selecting a Statistical Test ➊ Select an appropriate statistical test for a study. ➊ Brady is testing to see if there is a relationship between gender and political affiliation (Republican, Democrat, or independent). ➊ Lauren measures 20 participants pulse rates before drinking a cup of coffee and again 30 minutes afterward.