Discussion questions:
Problem 1: Heights of Females* (see end of document for more information)
Heights of females are known to follow a normal distribution with mean 64.5 inches and standard deviation 2.8 inches. Based on this information, answer the following questions.
a) Find the probability that a randomly selected female is taller than 67 inches. Draw a picture, shade area, standardize, and use Table 2 to obtain this probability. Please take a picture of your hand drawn sketch and upload it to your Word document. Verify your answer using a StatCrunch normal graph. Copy that image into your document as well.
b) Find the proportion of females who are between 60 and 62 inches tall (use only a StatCrunch graph).
c) Find the maximum height that would put a female in the bottom 4% of all female heights. Draw a picture, shade area, standardize, and use Table 2 to obtain this probability. Please take a picture of your hand drawn sketch and upload it to your Word document. Verify your answer using a StatCrunch normal graph. Copy that image into your document as well.
Problem 2: Grading on a Bell Curve
The mean and standard deviation of last semester's Exam 1 scores were 75.43 and 16.74 respectively. Imagine I wanted to define letter grades using a normal distribution (assuming this is appropriate). One possible grade distribution could be that the lowest 5% of students to earn Fs, the next 10% earn Ds, the next 35% earn Cs, the next 25% earn Bs, and the rest of the students earn As. Provide the Exam 1 scores that would separate these letter grades using the normal distribution. Your answer needs to be five normal graphs in StatCrunch where each shaded area on each graph shows the range of scores needed to earn the specific letter grade. Hint: the graphs for A and F will only have one value as a cutoff score whereas the graphs for a D, C, and B will have two cutoff scores (a low and a high score for that particular grade).
Problem 3: Common Last Names
The Census Bureau says that the 10 most common last names in the United States are (in order) Smith, Johnson, Williams, Jones, Brown, Davis, Miller, Wilson, Moore, and Taylor. These names account for 5.6% of all US residents. Consider our class as a random sample of 18 individuals.
a) Check if this situation fits the binomial setting.
b) Assuming it does, build the probability distribution as a table in StatCrunch. For your probability distribution table, since the probabilities will be very close to 0 after about X = 7, you may cut off the table after 7.
c) Find the probability that exactly 2 individuals have one of those last names. Provide a StatCrunch binomial graph displaying your answer.
d) Calculate the probability that at least one individual has one of those last names. Again, provide a StatCrunch binomial graph displaying your answer.
e) Calculate the probability that between 3 and 5 individuals (inclusive) have one of those last names. Use the probability distribution to answer this question (then verify it with a StatCrunch graph.
f) Find the mean and standard deviation of this probability distribution (you may use the binomial mean and standard deviation formulas in the notes).
Problem 4: Building a Sampling Distribution
Use my StatCrunch data set to build the sampling distribution of the sample proportion of college student's approval rating as we did in class (the data set is titled Opinions of 35,000 College Students Concerning the President and is posted in our StatCrunch group). This time, take 10,000 samples of size 50 and choose to display the sample proportion from each of the 10,000 samples. Give your computer a chance to collect the data; this may take up to 3 minutes. Be patient.
a) Graph the results in a histogram and discuss the shape, center, and spread.
b) Check the conditions of the central limit theorem and use the theorem to define the sampling distribution's shape, mean, and standard deviation and compare it to your histogram.
c) Lastly, for the defined sampling distribution, provide the probability that in a random sample of size 50, more than the majority supports the president.
Problem 5: Got Milk
according to the U.S. Department of Agriculture, 58.8% of males between 20 and 39 years old consume the minimum daily requirement of calcium. After an aggressive "Got milk" advertising campaign, the USDA conducted a survey of 55 randomly selected males between the ages of 20 and 39 and found that 36 of them consume the recommended daily allowance of calcium.
a) Construct a 99% confidence interval for the above data. Show your work using the formulas and verify your work using StatCrunch.
b) Interpret this confidence interval as we learned in class.
Note: To add formulas to a Word document, go above to Insert à object à Microsoft Equation 3.0. You can also copy and paste the following formulas when you need them (double click on the formula to replace the letters with numbers).