Find the Probability of at Least One Head

Exercises

Question 1

The probability distribution for the number of heads in 4 tosses of a coin is given by (Picture will be included in file)

Find the probability of at least one head.

Question 2

I roll a pair of fair dice and compute the number of spots on the two sides facing up. Denote this total by X. The probability distribution of X is

X / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10 / 11 / 12
P(X) / 1/36 / 2/36 / 3/36 / 4/36 / 5/36 / 6/36 / 5/36 / 4/36 / 3/36 / 2/36 / 1/36

Find the probability that X is at least 7.

Question 3

A random variable is

a.  a hypothetical list of the possible outcomes of a random phenomenon.

b.  any phenomenon in which outcomes are equally likely.

c.  any number that changes in a predictable way in the long run.

d.  a variable whose value is a numerical outcome of a random phenomenon.

Question 4

The density curve for a continuous random variable X has which of the following properties?

a.  The probability of any event is the area under the density curve and above the values of X that make up the event.

b.  The total area under the density curve for X must be exactly 1.

c.  The probability of any event of the form X = constant is 0.

d.  d. All of the above.

Question 5

The probability distribution for the number of heads X in 4 tosses of a coin is given by the table. (Picture included in files)

What is the mean number of heads?

Question 6

Red lights: A commuter must pass through five traffic lights on her way to work and will have to stop at each one that is red. She estimates the probability model for the number of red lights she hits, as shown below.

(Picture included in files: will say number 24)

a. How many red lights should she expect to hit each day?

b. What’s the standard deviation?

Question 7

Kittens: In a litter of seven kittens, three are female. You pick two kittens at random.

a. Create a probability model for the number of male kittens you get.

b. What’s the expected number of males?

c. What’s the standard deviation?

Question 8

Random variables: Given independent random variables with means and standard deviations as show, find the mean and standard deviation of: (Picture included in file, will say number 34)

a.  X – 20

b.  0.5Y

c.  X + 2Y

d.  X-Y

e.  Y1+Y2

Question 9

Each of the following situations is a two-way study design. For each case, identify the response variable and both factors, and state the number of levels for each factor (I and J) and the total number of observations (N).

a.  A study of smoking classifies subjects as nonsmokers, moderate smokers, or heavy smokers. Samples of 80 men and 80 women are drawn from each group. Each person reports the number of hours of sleep he or she gets on a typical night.

b.  The strength of concrete depends upon the formula used to prepare it. An experiment compares six different mixtures. Nine specimens of concrete are poured from each mixture. Three of these specimens are subjected to 0 cycles of freezing and thawing, three are subjected to 100 cycles, and three specimens are subjected to 500 cycles. The strength of each specimen is then measured.

c.  Four methods for teaching sign language are to be compared. Sixteen students in special education and sixteen students majoring in other areas are the subjects for study. Within each group they are randomly assigned to the methods. Scores on a final exam are compared.

Question 10

A two-way ANOVA model was used to analyze an experiment with two levels of one factor(Factor A), three levels of a second factor (Factor B), and 6 observations per treatment combination.

a.  For each of the main effects and the interaction, give the degrees of freedom (DF) for the corresponding F statistic.

b.  Find the value that each of these F statistics must exceed for the result to be significant at the 5% level.

c.  Answer part (b) for the 1% level.

Question 11

(Picture of problem included in files and will say number 28) Exercise 13.28a. Compute the mean and standard deviation for each type of pot. Question 4. Exercise 13.28c. Give the ANOVA table and your conclusions regarding the hypotheses about main effects and interaction. Question 5. Fill in the ANOVA table and give your conclusions regarding the hypotheses about main effects and interaction.

Source / DF / SS / MS / F / P
A(Gender) / 1 / 62
B(Gender) / 1 / 1,599
AB(Interact)
Error / 13,633
Total / 599 / 15,458

Question 12

Contacts: Assume that 30% of students at a university wear contact lenses. Can we apply the normal model to approximate the distribution of the sample proportion?

a. We randomly pick 100 students. Let p represent the proportion of students in this sample who wear contacts. What’s the appropriate model for the distribution of p? Specify the name of the distribution, the mean, and the standard deviation. Be sure to verify that the conditions are met.

b. What is the mean of the sample proportion distribution?

c. What is the standard deviation of the sample proportion distribution?

d. What is the approximate probability that more than one third of this sample wear contacts?

Question 13

Rainfall: Statistics from Cornell’s Northeast Regional Climate Center indicate that Ithaca, NY, gets an average of 35.4” of rain each year, with a standard deviation of 4.2”. Assume that a Normal model applies.

a. During what percentage of years does Ithaca get more than 40” of rain?

b. Less than how much rain falls in the driest 20% of all years?

What is the mean of the sampling distribution?

c. A Cornell university student is in Ithaca for four years. Let y represent the mean amount of rain for those 4 years. Describe the sampling distribution model of this sample mean, y.

What is the standard deviation of the sampling distribution?

d. What’s the probability that those 4 years average less than 30” of rain?

Question 14

The amount that households pay service providers for access to the Internet varies quite a bit, but the mean monthly fee is $28 and the standard deviation is $10. The distribution is not normal: many households pay about $25 for unlimited dial-up access, but many pay more for broadband connections. A sample survey asks an SRS of 500 householders with Internet access how much they pay. What is the probability that the average fee paid by the sample households exceeds $29?

Question 15

A May 2007 Gallup Poll found that only 11% of a random sample of 1003 adults approved of attempts to clone a human.

a.  Find the margin of error for this poll if we want 95% confidence in our estimate.

b.  If we only need to be 90% confident, will the margin of error be larger or smaller.

c.  Find the margin of error for this poll if we want 90% confidence in our estimate.

d.  In general, if all aspects of the situation remain the same, would smaller samples produce smaller or larger margins of error?

Question 16

Gambling: A city ballot includes a local initiative that would legalize gambling. The issue is hotly contested, and two groups decide to conduct polls to predict the outcome. The local newspaper finds that 53% of 1200 randomly selected voters plant to vote “yes”, while a college Statistics class finds 54% of 450 randomly selected voters in support. Both groups will create 95% confidence intervals.

a.  Without finding the confidence intervals which one will have the larger margin of error.

b.  Find the local newspaper's confidence interval.

c.  Find the college class’ confidence interval.

d.  Which group concludes that the outcome is too close to call? Why?

Question 17

Hiring: In preparing a report on the economy, we need to estimate the percentage of businesses that plan to hire additional employees in the next 60 days.

a.  How many randomly selected employers must we contact in order to create an estimate in which we are 98% confident with a margin of error of 5%? Assume the percentage of businesses equals to 50%.

b.  How many randomly selected employers must we contact in order to create an estimate in which we are 98% confident with a margin of error of 3%? Assume the percentage of businesses equals to 50%. What sample size will suffice?

c.  How many randomly selected employers must we contact in order to create an estimate in which we are 98% confident with a margin of error of 1%? Assume the percentage of businesses equals to 50%. Why might it not be worth the effort to try to get an interval with a margin of error of only 1%?

Question 18

A poll of 811 adults aged 18 or older asked about purchases that they intended to make for the upcoming holiday season. One of the questions asked about what kind of gift they intended to buy for the person on whom they would spend the most. Clothing was the first choice of 487 people. Give a 99% confidence interval for the proportion of people in this population who intend to buy clothing as their first choice. (Use the sample proportion p(hat)=X/n for estimation)

Question 19

In a survey of 1280 student loan borrowers, there were 1050 borrowers whose total debt was $10,000 or more. Of these, 192 left school without completing a degree. Consider the population to be borrowers whose total debt was $10,000 or more.
Give the left endpoint (lower boundary) of a 95% confidence interval for the proportion of borrowers who left school without completing a degree in this population.
Use the sample proportion p(hat)=X/n for estimation

Question 20

For the problem in question 19 give the right endpoint (upper boundary) of a 95% confidence interval for the proportion.

Question 21

The P-value of a test of a null hypothesis is

a.  the probability, assuming the null hypothesis is true, that the test statistic will take a value at least as extreme as that actually observed.

b.  the probability, assuming the null hypothesis is false, that the test statistic will take a value at least as extreme as that actually observed.

c.  the probability that the null hypothesis is true.

d.  the probability that the null hypothesis is false.

Question 22

P-value for a significance test is 0.022. Do you reject the null hypothesis at significant level 0.05.

Question 23

More hypotheses: Write the null and alternative hypotheses you would use to test each situation.

In each situation decide is the test left-sided, right-sided or two-sided.

a.  In the 1950s, only about 40% of high school graduates went onto college. Has the percentage changed?

b.  Twenty percent of cars of a certain model have needed costly transmission work after being driven between 50,000 and 100,000 miles. The manufacturer hopes that a redesign of a transmission component has solved this problem.

c.  We field test a new flavor soft drink, planning to market it only if we are sure that over 60% of the people like the flavor.

Question 24

Dice: The seller of a loaded die claims that it will favor the outcome of 6. We don’t believe that claim, and roll the die 200 times to test an appropriate hypothesis. Our P-value turns out to be 0.03. Which conclusion is appropriate? Explain.

a.  There’s a 3% chance that the die is fair.

b.  There’s a 97% chance that the die is fair.

c.  There’s a 3% chance that a loaded die could randomly produce the results we observed, so it’s reasonable to conclude that the die is fair.

d.  There’s a 3% chance that a fair die could randomly produce the results we observed, so it’s reasonable to conclude that the die is loaded.

Question 25

Obesity 2008: In 2008, the CDC reported that 34% of adults in the United States are obese. A county health service planning a new awareness campaign polls a random sample of 750 adults living there. In this sample, 228 people were found to be obese based on their answers to a health questionnaire. Do these responses provide strong evidence that the 34% figure is not accurate for this region? Correct the mistakes you find in a student’s attempt to test an appropriate hypothesis. (Last picture included in files with the formulas/equations)

Question 26

Educated mothers: The national center for education statistics monitors many aspects of elementary and secondary education nationwide. Their 1996 numbers are often used as a baseline to assess changes. In 1996, 31% of students reported that their mothers had graduated from college. In 2000, responses from 8368 students found that this figure had grown to 32%. Is this evidence of a change in education level among mothers? Set up null and alternative hypotheses about the population proportion P.

a.  Write appropriate hypotheses.

b.  Check the assumptions and conditions. Are the assumptions and conditions necessary for inference satisfied?

c.  Perform the test and find the P-value.

d.  State your conclusion using 5% level of significance.
Hint: If P-value is less than or equal to 0.05 reject Ho.

e.  Do you think this difference is meaningful? Explain.

Question 27

Of the 500 respondents in the Christmas tree market, 44% had no children at home and 56% had at least one child at home. The corresponding figures for the most recent census are 48% with no children and 52% with. at least one child. Test the null hypothesis that the telephone survey technique has a probability of selecting a household with no children that is equal to the value obtained by the census.
Set up null and alternative hypotheses about proportion P of households with no children.

Question 28

For the problem in question 24 compute the value of the test statistic z. Use the sample proportion 0.44.

Question 29

For the problem in question 24 give the P-value of the test.

Question 30

What do you conclude for the problem in question 24 at significance level = 0.05?