Statistics Problem Set III Final Exam Problem Set

The following is known about test scores obtained by 30 students in a Statistics course: Sx = 2220 Sx2 = 173586 The mean = 𝑥 𝑥 = 𝑥 𝑁 𝑥 = 2220 30 = 74 a. Approximately how many people obtained a score between 80 and 90? The only way to answer questions about proportions contained between scores is to use z-scores. This is an “area under the curve” z-score problem. We need the mean and standard deviation.

The standard deviation = s The following is known about test scores obtained by 30 students in a Statistics course: Sx = 2220 Sx2 = 173586 The standard deviation = s s = 𝑆𝑆 𝑁−1 SS = 𝑥 2 − 𝑥 2 𝑁 SS =173586− 2220 2 30 SS =173586−164280 SS =9306 a. Approximately how many people obtained a score between 80 and 90? The only way to answer questions about proportions contained between scores is to use z-scores. This is an “area under the curve” z-score problem. We need the mean and standard deviation.

The standard deviation = s The following is known about test scores obtained by 30 students in a Statistics course: Sx = 2220 Sx2 = 173586 The standard deviation = s s = 𝑆𝑆 𝑁−1 s = 9306 29 s = 320.90 s =17.91 a. Approximately how many people obtained a score between 80 and 90? The only way to answer questions about proportions contained between scores is to use z-scores. This is an “area under the curve” z-score problem. We need the mean and standard deviation.

The following is known about test scores obtained by 30 students in a Statistics course: Sx = 2220 Sx2 = 173586 Calculating Z-scores z = x− 𝑥 𝑠 Z-score for 90 = 90−74 17.91 = .89 Z-score for 80 = 80−74 17.91 = .34 𝑥 =74 s =17.91 a. Approximately how many people obtained a score between 80 and 90? The only way to answer questions about proportions contained between scores is to use z-scores. This is an “area under the curve” z-score problem. What type of area is this? It is neither an area B or an area C in itself. We must combine areas. We must subtract Area Bs. First, calculate z-scores, then find Area B: 74 80 90

Area B for z-score of .34 = .1331 Area B for z-score of .89 = .3133 .34 .89

a. Approximately how many people obtained a score between 80 and 90? The following is known about test scores obtained by 30 students in a Statistics course: Sx = 2220 Sx2 = 173586 𝑥 =74 s =17.91 a. Approximately how many people obtained a score between 80 and 90? The only way to answer questions about proportions contained between scores is to use z-scores. This is an “area under the curve” z-score problem. What type of area is this? It is neither an area B or an area C in itself. We must combine areas. We must subtract Area Bs. First, calculate z-scores, then find Area B: Z-score for 90: .89  Area B = .3133 Z-score for 80: .34  Area B = .1331 .3133-.1331 = .1802 30 x .18 = 5.4 people 74 80 90

b. If lower than 65 is a failing grade, how many people failed? The following is known about test scores obtained by 30 students in a Statistics course: Sx = 2220 Sx2 = 173586 𝑥 =74 s =17.91 b. If lower than 65 is a failing grade, how many people failed? We must first calculate the z-score for a raw score of 65: 𝑧= 𝑥− 𝑥 𝑠 𝑧= 65−74 17.91 = -.50 Since this is an Area C, we look up the Area C value for a z-score of -.50 (remember no negatives in the table!) 65 74

Area C for z-score of -.50= .3085 -.50

b. If lower than 65 is a failing grade, how many people failed? The following is known about test scores obtained by 30 students in a Statistics course: Sx = 2220 Sx2 = 173586 𝑥 =74 s =17.91 b. If lower than 65 is a failing grade, how many people failed? We must first calculate the z-score for a raw score of 65: 𝑧= 𝑥− 𝑥 𝑠 𝑧= 65−74 17.91 = -.50 Since this is an Area C, we look up the Area C value for a z-score of -.50 (remember no negatives in the table!) Area C = .3085 30 x .3085 = 9.255, or 9 people. 65 74

The following is known about test scores obtained by 30 students in a Statistics course: Sx = 2220 Sx2 = 173586 𝑥 =74 s =17.91 c. If only the top 10% of scores receive As, what is the lowest score required in order to still get an A? We know this is an Area C, and we know that its area is .10, or 10%. We do NOT know what the raw score is, so we use the table but we will start with Area C this time. 74

Area C closest to .1000 without going over. Z-score that has an area C closest to 10% is 1.29.

The following is known about test scores obtained by 30 students in a Statistics course: Sx = 2220 Sx2 = 173586 𝑥 =74 s =17.91 c. If only the top 10% of scores receive As, what is the lowest score required in order to still get an A? We know this is an Area C, and we know that its area is .10, or 10%. We do NOT know what the raw score is, so we use the table but we will start with Area C this time. z-score which has an Area C closest to .10 is 1.29 Did we answer the question yet? Must convert 1.29 to a raw score: 𝒙= 𝒙 +𝒛 𝒔 𝒙=𝟕𝟒+𝟏.𝟐𝟗 𝟏𝟕.𝟗𝟏 𝒙=𝟕𝟒+𝟐𝟑.𝟏𝟎 𝒙=𝟗𝟕.𝟏𝟎 74

d. What proportion of individuals obtained a score between 65 and 80? The following is known about test scores obtained by 30 students in a Statistics course: Sx = 2220 Sx2 = 173586 𝑥 =74 s =17.91 d. What proportion of individuals obtained a score between 65 and 80? This is going to be a combination of areas: we must ADD area Bs: First, find the z-score for 65 and 80: z-score for 65: -.50 z-score for 80: .34 Then, look up their area Bs: Area B for -.50 = .1915 Area B for .34 = .1331 Then, add them: .1915 + .1331 = .3246 or 32.46% of people 65 80 74

Predict the GPA of a student whose pupil diameter is 8.85mm: The following is known about 10 students’ pupil diameters (in mm) and their GPA: r = .80 Predict the GPA of a student whose pupil diameter is 8.85mm: b. Predict the pupil diameter of a 4.0 GPA student:   𝐩𝐮𝐩𝐢𝐥 GPA 𝑴𝒆𝒂𝒏 8 2.4 s 1.7 .4 x y Find slope: Find intercept: Equation of regression line:   𝐩𝐮𝐩𝐢𝐥 GPA 𝒎𝒆𝒂𝒏 8 2.4 s 1.7 .4 𝑏=𝑟 𝑠 𝑦 𝑠 𝑥 𝑎= 𝑦 −𝑏 𝑥 𝑦=𝑏𝑥+𝑎 𝑦=.19𝑥+.88 𝑎=2.4−.19(8) 𝑏=.80 .4 1.7 𝑦=.19(8.85)+.88 𝑎=.88 𝑦=2.56 𝑏=.19 y x Find slope: Find intercept: Equation of regression line:   𝐩𝐮𝐩𝐢𝐥 GPA 𝒎𝒆𝒂𝒏 8 2.4 s 1.7 .4 𝑏=𝑟 𝑠 𝑦 𝑠 𝑥 𝑎= 𝑦 −𝑏 𝑥 𝑦=𝑏𝑥+𝑎 𝑎=8−3.4(2.4) 𝑦=3.4𝑥−.16 𝑏=.80 1.7 .4 𝑎=−.16 𝑦=3.4 4 −.16 𝑦=13.44 𝑏=3.4

You conduct a study of the effects of the new drug -glutraine on human memory. You test a sample of 30 participants in your FDA approved laboratory on standard memory tests that provides a performance score for each participant on a scale of 1 to 10. Where the score indicates the number of items recalled (out of a total ten words). You divide 30 particpants into three groups of 10. The first group receives and sugar pill one hour before tests; the second group receives a small dose of a-glutraine; and the third group receives a large dose of a-glutraine. Once 1 hour after an a-glutraine pill has been swallowed by each participant the participant is given ten minutes to memorize a 10-word written passage from a novel. One hour later each subject is tested on his or her memory performance (use alpha=.05).

You conduct a study of the effects of the new drug -glutraine on human memory. You test a sample of 30 participants in your FDA approved laboratory on standard memory tests that provides a performance score for each participant on a scale of 1 to 10. Where the score indicates the number of items recalled (out of a total ten words). You divide 30 particpants into three groups of 10. The first group receives and sugar pill one hour before tests; the second group receives a small dose of a-glutraine; and the third group receives a large dose of a-glutraine. Once 1 hour after an a-glutraine pill has been swallowed by each participant the participant is given ten minutes to memorize a 10-word written passage from a novel. One hour later each subject is tested on his or her memory performance (use alpha=.05).

I would like to evaluate whether my cat is psychic I would like to evaluate whether my cat is psychic. I present her with 4 treats simultaneously where each treat is accompanied by a photo of a racehorse. I then record which treat she eats first and use this information to place a bet on a horse race in which these 4 horses are competing. I do this 14 times. a. What is the probability she will guess the winner of all 14 races? b. What is the probability she will guess the winner of at least 10 of the 14 races? c. What is the probability she will guess the winner of at least 8 races out of 14 if 5 horses were competing (and I gave her 5 treats to choose from)? d. What is the probability she will guess the winner of all 14 races if only two horses compete (and I only put two treats in front of her)? .0000 .0003 .0023 .0001

The following is known about a set of 5 sample test scores: Sx = 34 Sx2 = 396 a. Find the Sum of Squares (SS): b. Find the standard deviation: c. Find the standard error: d. Find the variance:

The following is known about a set of 5 sample test scores: Sx = 34 Find the Sum of Squares (SS): b. Find the standard deviation: c. Find the standard error: d. Find the variance: 𝑆𝑆= 𝑥 2 − 𝑥 2 𝑁 𝑆𝑆=396 − 34 2 5 𝑆𝑆=396 −231.2 𝑆𝑆=164.8

The following is known about a set of 5 sample test scores: Sx = 34 a. Find the Sum of Squares (SS): 164.8 b. Find the standard deviation: c. Find the standard error: d. Find the variance: 𝑠= 𝑆𝑆 𝑁−1 𝑠= 164.8 4 𝑠= 41.2 𝑠=6.42

The following is known about a set of 5 sample test scores: Sx = 34 a. Find the Sum of Squares (SS): 164.8 b. Find the standard deviation: 6.42 c. Find the standard error: d. Find the variance: 𝑠 𝑥 = 𝑠 𝑁 𝑠 𝑥 = 6.42 5 𝑠 𝑥 = 6.42 2.24 𝑠 𝑥 =2.87

The following is known about a set of 5 sample test scores: Sx = 34 a. Find the Sum of Squares (SS): 164.8 b. Find the standard deviation: 6.42 c. Find the standard error: 2.87 d. Find the variance: 𝑠 2 = 𝑆𝑆 𝑁−1 𝑠 2 = 164.8 4 𝑠 2 =41.2

The following is known about a set of 5 sample test scores: Sx = 34 a. Find the Sum of Squares (SS): 164.8 b. Find the standard deviation: 6.42 c. Find the standard error: 2.87 d. Find the variance: 41.2 Another way to find the variance would be to square the standard deviation: 𝑠 2 = 6.42 2 𝑠 2 =41.22

What is the appropriate test? State the null hypothesis: The average household in America changes light bulbs about 5 times a year. General Electrics is unveiling a new light bulb which they claim will last longer than the average light bulb, and so it will have to be replaced less frequently. A research lab buys enough of these new light bulbs to fit 10 American households. After one year of use, they ask those families to report how many times they changed their bulbs. Below is the data. Test the hypothesis that these new bulbs last longer than the average light bulb (use a = .05). x = {1, 1, 5, 1, 8, 1, 1, 2, 1, 9} What is the appropriate test? State the null hypothesis: State the alternative hypothesis: Find the critical value: Calculate the obtained statistic: Make a decision: What does your decision mean? One-sample t-test The new bulbs last longer than the average bulb. m = 5 N = 10 The new bulbs do not last longer than the average bulb. a = .05 One-tailed, a=.05, 9 df 𝑥 =3 𝑠=3.16

The average household in America changes light bulbs about 5 times a year. General Electrics is unveiling a new light bulb which they claim will last longer than the average light bulb, and so it will have to be replaced less frequently. A research lab buys enough of these new light bulbs to fit 10 American households. After one year of use, they ask those families to report how many times they changed their bulbs. Below is the data. Test the hypothesis that these new bulbs last longer than the average light bulb (use a = .05). If the new bulbs last longer, we should have to change them LESS. 5 So tcrit = - 1.83

The average household in America changes light bulbs about 5 times a year. General Electrics is unveiling a new light bulb which they claim will last longer than the average light bulb, and so it will have to be replaced less frequently. A research lab buys enough of these new light bulbs to fit 10 American households. After one year of use, they ask those families to report how many times they changed their bulbs. Below is the data. Test the hypothesis that these new bulbs last longer than the average light bulb (use a = .05). x = {1, 1, 5, 1, 8, 1, 1, 2, 1, 9} What is the appropriate test? State the null hypothesis: State the alternative hypothesis: Find the critical value: Calculate the obtained statistic: Make a decision: What does your decision mean? One-sample t-test The new bulbs last longer than the average bulb. m = 5 N = 10 The new bulbs do not last longer than the average bulb. a = .05 One-tailed, a=.05, 9 df = -1.83 𝑥 =3 𝑠=3.16 𝑡 𝑜𝑏𝑡 = 𝑥 − 𝜇 𝑥 𝑠 𝑥 = 3−5 1 =−2 𝜇 𝑥 =5 -1.83 𝑠 𝑥 = 𝑠 𝑁 = 3.16 10 =1 Reject the null hypothesis. The new bulbs last longer than the average bulb.