Let A, B, and C be independent random variables, uniformly distributed over [0,6],[0,11], and [0,2] respectively. What is the probability that both roots of the equation Ax^2+Bx+C=0 are real?

Someone claims that the mean number of sick days that employees in New Jersey take per year is 5.3. To look into that claim, you take a representative sample of 78 employees in New Jersey and find that the mean number of sick days is 5.5 in the sample. The population standard deviation is 1.6.

Part (a)   

Given that the sample mean is different from the claimed population mean, does that show that the claim in H₀ is false? Explain your answer.

Part (b)

Carry out a hypothesis test for the claim above (with α = 0.05) using the 6-step procedure.

Part (c)   

Carry out a hypothesis test for the claim above (with α = 0.05) using the p-value method.

Professor Fair believes that extra time does not improve grades on exams. He randomly divided a group of 300 students into two groups and gave them all the same test. One group had exactly 1 hour in which to finish the test, and the other group could stay as long as desired. The results are shown in the following table. Test at the 0.01 level of significance that time to complete a test and test results are independent.

(i) Give the value of the level of significance.


State the null and alternate hypotheses.

H₀: Time to take a test and test score are not independent.
H₁: Time to take a test and test score are independent. H₀: The distributions for a timed test and an unlimited test are the same.
H₁: The distributions for a timed test and an unlimited test are different.     H₀: The distributions for a timed test and an unlimited test are different.
H₁: The distributions for a timed test and an unlimited test are the same. H₀: Time to take a test and test score are independent.
H₁: Time to take a test and test score are not independent.

(ii) Find the sample test statistic. (Round your answer to two decimal places.)


(iii) Find or estimate the P-value of the sample test statistic.

P-value > 0.100 0.050 < P-value < 0.100     0.025 < P-value < 0.050 0.010 < P-value < 0.025 0.005 < P-value < 0.010 P-value < 0.005

(iv) Conclude the test.

Since the P-value < α, we reject the null hypothesis. Since the P-value is ≥ α, we do not reject the null hypothesis.     Since the P-value < α, we do not reject the null hypothesis. Since the P-value ≥ α, we reject the null hypothesis.

(v) Interpret the conclusion in the context of the application.

At the 1% level of significance, there is insufficient evidence to claim that time to do a test and test results are not independent. At the 1% level of significance, there is sufficient evidence to claim that time to do a test and test results are not independent.    

"A random survey of 927 adults in California found that 63% of them say they are likely to sleep when they stay home sick."

e. Construct a 95% confidence interval for p. (Be sure to follow the whole process).

f. Is it plausible to say that 70% of all California adults would sleep when they stay home sick?

g. Perform a hypothesis test to determine if more than 60% of adults would sleep when they stay home sick.

h. If we made an error here, then would it be a Type I or a Type II error?

The director of a Masters of Public Administration Program is preparing a brochure to promote the

program. She would like to include in the brochure the average grade point average (GPA) of first-

year students in the program, but since time is pressing she decides to estimate this figure with a

sample of ten students (GPAs are normally distributed). The GPAs are listed below. What is the best

estimate of the average GPA for all first-year students? With 95% confidence, can the director

conclude the average GPAs for first-year students is a B or better (3.0 on a 4.0 scale)?

2.8

3.6

3.4

2.5

2.2

2.6

4.0

3.1

2.7

3.5

In a recent issue of Consumer Reports, Consumers Union reported on their investigation of bacterial contamination in packages of name brand chicken sold in supermarkets.

Packages of Tyson and Perdue chicken were purchased. Laboratory tests found campylobacter contamination in 35 of the 75 Tyson packages and 22 of the 75 Perdue packages.

Question 1. Find 90% confidence intervals for the proportion of Tyson packages with contamination and the proportion of Perdue packages with contamination (use 3 decimal places in your answers).

lower bound of Tyson interval
upper bound of Tyson interval
lower bound of Perdue interval
upper bound of Perdue interval

Question 2. The confidence intervals in question 1 overlap. What does this suggest about the difference in the proportion of Tyson and Perdue packages that have bacterial contamination? One submission only; no exceptions

The overlap suggests that there is no significant difference in the proportions of packages of Tyson and Perdue chicken with bacterial contamination.

Even though there is overlap, Tyson's sample proportion is higher than Perdue's so clearly Tyson has the greater true proportion of contaminated chicken.    

Question 3. Find the 90% confidence interval for the difference in the proportions of Tyson and Perdue chicken packages that have bacterial contamination (use 3 decimal places in your answers).

lower bound of confidence interval
upper bound of confidence interval

Question 4. What does this interval suggest about the difference in the proportions of Tyson and Perdue chicken packages with bacterial contamination? One submission only; no exceptions

Tyson's sample proportion is higher than Perdue's so clearly Tyson has the greater true proportion of contaminated chicken.

Natural sampling variation is the only reason that Tyson appears to have a higher proportion of packages with bacterial contamination.    

We are 90% confident that the interval in question 3 captures the true difference in proportions, so it appears that Tyson chicken has a greater proportion of packages with bacterial contamination than Perdue chicken.

Question 5. The results in questions 2 and 4 seem contradictory. Which method is correct: doing two-sample inference, or doing one-sample inference twice? One submission only; no exceptions

one-sample inference twice

two-sample inference    

Question 6. Why don't the results agree? 2 submission only; no exceptions

Different methods were used in the two samples to detect bacterial contamination.

The one- and two-sample procedures for analyzing the data are equivalent; the results differ in this problem only because of natural sampling variation.    

If you attempt to use two confidence intervals to assess a difference between proportions, you are adding standard deviations. But it's the variances that add, not the standard deviations. The two-sample difference-of-proportions procedure takes this into account.

Tyson chicken is sold in less sanitary supermarkets.

Suppose a hypertension trial is mounted and 18 participants are randomly assigned to one of the comparison treatments. Each participant takes the assigned medication and their systolic blood pressure (SBP) is recorded after 6 months on the assigned treatment. Is there a difference in mean SBP among the three treatment groups at the 5% significance level? The data are as follows.  

What is total variance (or as what it's called in ANOVA, "MS_total")?

A. 13.8

B.   189.6

C. 3222.9

D. 179.1

A child development specialist is interested in learning if a new learning program increases students’ memory. 15 Subjects learned a list of 50 words. Learning performance was measured using a recall test. Students were initially tested and then tested again after using the new program. Below is the number of words remembered by each student.

Student #                    Score 1                  Score 2

         1                              24                           26          

                2                              17                           24

                3                              32                           31

                4                              14                           17

                5                              16                           17

                6                              22                           25

                7                              26                           25

                8                              19                           24

                9                              19                           22

               10                             22                           23

              11                          21                           26

              12                             25                           28

             13                             16                           19

            14                         24                           23

            15                           18                           22

Did the learning program significantly improve the student’s ability to recall words? Report standard error of means, df, obtained and critical t, and whether you would accept or reject the null hypothesis.

Iconic memory is a type of memory that holds visual information for about half a second (0.5 seconds). To demonstrate this type of memory, participants were shown three rows of four letters for 50 milliseconds. They were then asked to recall as many letters as possible, with a 0-, 0.5-, or 1.0-second delay before responding. Researchers hypothesized that longer delays would result in poorer recall. The number of letters correctly recalled is given in the table.

(a) Complete the F-table. (Round your values for MS and F to two decimal places.)

(b) Compute Tukey's HSD post hoc test and interpret the results. (Assume alpha equal to 0.05. Round your answer to two decimal places.)

The critical value is_______ for each pairwise comparison.

Which of the comparisons had significant differences? (Select all that apply.)

A.) The null hypothesis of no difference should be retained because none of the pairwise comparisons demonstrate a significant difference.

B.) Recall following no delay was significantly different from recall following a half second delay.

C.) Recall following a half second delay was significantly different from recall following a one second delay.

D.) Recall following no delay was significantly different from recall following a one second delay.

1. People claim that women say more words per day than men. Estimates claim that a woman uses roughly 20,000 words per day, while a man uses approximately 7,000. To investigate this, a researcher recorded conversations of male college students over a 5-day period. The results are as follows:

The researcher believes that many use more than 7,000 words per day.

1. State the problem in your own words.
2. Create a plan for testing the researcher’s claim. Be sure to state the null and alternate hypotheses.
3. Carry out the appropriate hypothesis test. Begin by finding the sample mean and standard deviation. Give the value of the t statistic and give the p-value (or an estimate).
4. Formulate the statistical conclusion in terms of the null hypothesis and a practical conclusion. You may compare the p-value to 0.05 to determine if the null hypothesis should be rejected. When the p-value is less than 0.05, we reject the null hypothesis. Otherwise, we fail to reject the null.

Many high school students take the SAT's twice; once in their Junior year and once in their Senior year. The Senior year scores (x) and associated Junior year scores (y) are given in the table below. This came from a random sample of 35 students. Use this data to test the claim that retaking the SAT increases the score on average by more than 25 points. Test this claim at the 0.10 significance level.

Are all colors equally likely for Milk Chocolate M&M's? Data collected from a bag of Milk Chocolate M&M's are provided.

Blue            Brown        Green          Orange       Red          Yellow

110                   47               52                 103            58                50

a. State the null and alternative hypotheses for testing if the colors are not all equally likely for Milk Chocolate M&M's.

b. If all colors are equally likely, how many candies of each color (in a bag of 420 candies) would we expect to see?

c. Is a chi-square test appropriate in this situation? Explain briefly.

d. How many degrees of freedom are there?

A) 2 B) 3 C) 4 D) 5

e. Calculate the chi-square test statistic. Report your answer with three decimal places.

f. Report the p-value for your test. What conclusion can be made about the color distribution for Milk Chocolate M&M's? Use a 5% significance level.

g. Which color contributes the most to the chi-square test statistic? For this color, is the observed count smaller or larger than the expected count?