Statistics Assignment: SAT and ACT Scores, Sampling Distribution, Hypothesis Testing

Question 1: SAT Scores Distribution and Scholarship Consideration

1.The distribution of SAT exam scores is approximately normal with  = 500 and  = 100.  For the population of students who have taken the SAT,

a.What proportion of students has SAT scores between 450 and 580?

b.What is the percentile rank of someone who scores a 650?

c.To be considered for a scholarship at State of Confusion College of University Studies, a student must be in at least the 40th percentile of all SAT test takers.  What is the lowest SAT score that will be considered for a scholarship?

2.As mentioned previously, the distribution of ACT scores is approximately normal with  = 20 and  = 5.

a.If you selected a random sample of n = 40 scores from this population, how much error would you expect between the sample mean and the actual population mean (i.e., what is the standard deviation of the sampling distribution σ  or the standard error)?  

b.If you selected a random sample of n = 400 scores, how much error would you expect between the sample mean and the actual population mean?

c.How much error would you expect for a sample of n = 4000 scores?

3.Indicate which of the following statements is not true?

a. The mean of the sampling distribution of sample means will always be equal to the population mean regardless of the shape of the distribution.

b. The standard error is always smaller than the standard deviation except when we take a sample of n = 1.

c. The Unit Normal Table can only be used when the distribution of scores in the population is approximately Normal.  

4.Using imaginary data from this class, we can say that the population distribution of scores is approximately Normal with a mean score (µ) on the midterm of 85.25 and a standard deviation (σ) of 6.4.  If I were to take a random sample of size n = 16, what is the probability that the sample mean is:  

A complete answer for a b and c will include an interpretive sentence relating the probability to the sample mean

a.Higher than 87? 

b.Between 83 and 89? 

c.Less than 82.05? 

d.Extra Credit 1 point.  Briefly show that we would be more likely to get a sample mean higher than 82 if my sample size (n) was 32 instead of 16 if the population mean and standard deviation remain the same.

5.(From Frankfort-Nachmias and Leon-Guerrero, 2002) For each of the following situations state the null and research hypotheses.  Pay attention to whether these should be constructed as a one –tailed or a two-tailed test.  

a.Researcher A is interested in finding out if the average income of elementary school teachers is different from the national average income for adults.  According to recent census information, the average income for adults age 25 and older is $40,000.

b.Researcher B believes that students in small liberal arts colleges attend more parties per month than students nationwide.  Previous research has shown that nationally, undergraduate students attend an average of 2.5 parties per month.

c.Researcher C thinks that stress (measured on an “interval” scale from 0-100, 0 being non-existent and 100 being extremely high) will be lower for adults who own dogs (or other pets) than for the general adult population.  The population mean (µ) of the general population is 50.

d.Make up your own research scenario where Researcher D will use a one-tailed hypothesis. No more than 50 words, but descriptive enough that it is clear what the hypothesis structure must be. Include the hypotheses that are implied by your research scenario.