Exam 1 PS 217, Spring 20031. Given that SAT math scores are normally distributed with m = 500 and s = 100, answer the following questions:[10 pts]a. If all the SAT math scores were transformed to z-scores, the distribution of z-scores would have parameters asfollows (fill in the box below each parameter):ms01b. If a person achieved an SAT math score of 535, what proportion (or percentage) of people would have higherSAT math scores? z =535 - 500100= .35, so .3632 (or 36%) would have SAT scores of 535 or higherc. What SAT math scores would determine the middle 50% of the distribution? (In other words,25% above and 25% below the mean.) z = ±.67 =x - 500100, so SAT scores between 433 and 567 would determine the middle 50% ofthe scoresd. What proportion (or percentage) of people would have SAT math scores between 550 and650? z =550 - 500100= 0.5 and z =650 - 500100= 1.5, so .3085 - .0668 = .2417 or 24.17%2. Suppose that you take a sample of n = 16 students and determine the GPA for each of them, as seen below:GPAGPA23.512.253.09.03.814.442.98.412.56.253.512.253.19.613.09.02.98.413.411.563.512.252.14.413.411.563.814.443.210.242.77.29Sum50.3161.37a. Estimate the parameters of the population from which the sample was drawn. [5 pts] M or X =ˆ m =50.316= 3.14 s2=ˆ s 2=SSdf=3.2415= .22SS = 161.37 -50.3216= 3.24 s =ˆ s = .22 = .47b. Test the hypothesis that the sample was drawn from a population with m = 3.0. [10 pts]H0: m = 3.0H1: m ≠ 3.0tCrit(15) = 2.131 sX or sM=.2216= .12 so tObt=3.14 - 3.12= 1.17Decision: Retain H0, because |tObt| < tCritConclude: Sample may have been drawn from population with m = 3.0 (lack of power)3. In the context of signal detection theory in the lab on z-scores, you learned about the possible outcomes of trials,as seen below: [5 pts]Type of TrialNoise OnlySignal + NoiseSignal PresentFalse AlarmHitParticipant ReportsNo Signal PresentCorrect RejectionMissIt should strike you that this table is very similar to the table used to illustrate the possibilities of decisions inhypothesis testing (e.g., Type I error, Type II error). Set up the table below so that it mirrors the table above in termsof H0, etc.Actual SituationH0 TrueH0 FalseReject H0Type I ErrorCorrect RejectionExperimenter'sDecisionRetain H0Correct RetentionType II Error4. As you saw in that lab, you can use z-scores to determine a measure of sensitivity (d'). Suppose that on arecognition memory test, a person got 80% hits and 10% false alarms. What d' would that person receive? [5 pts]80% Hits would give you a z-score of -0.84 (in the body, Col. B). 10% False Alarms wouldgive you a z-score of 1.28 (in the tail, Col. C). Thus, d’ = 2.12 (1.28 + 0.84).5. As you know, IQ scores are normally distributed with m = 100 and s = 15. Use that information to address thefollowing questions. [10 pts]a. What proportion (or percentage) of people have IQ scores between 100 and 115? z =115 - 10015= 1 and z =100 - 10015= 0, so the proportion is .3413 (or 34.13%)b. If you took a sample of n = 25, what proportion (or percentage) of such samples would have means between 100and 115? sX or sM=1525= 3z =115 - 1003= 5 and z =100 - 1003= 0Thus, roughly 50% of the samples would have means between 100 and 115.c. Briefly explain why you might have gotten a different answer for a and b above.In a, the distribution is comprised of raw scores, so it would have a standard deviation of15. In b, the distribution is comprised of sample means, so it would have a standarddeviation (standard error) of 3. In general, the sampling distribution of the mean will haveless variability than the population from which the samples were drawn.6. A sample of n = 36 quiz scores were obtained from a statistics class. The StatView analysis conducted on thesedata is seen below. [5 pts]8.222 35 .732 .4692Mean DF t-Value P-ValueQuiz ScoresOne Sample t-testHypothesized Mean = 8a. State the null and alternative hypothesis.H0: m = 8H1: m ≠ 8b. What should you conclude about H0?Because the P-Value is greater than .05, I would retain H0.c. What would you say about the power of this analysis?Given the failure to reject H0, there appears to be too little power in this
View Full Document
Unlocking...