At wind speed above 1000 cm/sec, significant sand-moving events begin to occur. Wind speeds below 1000 cm/sec deposit sand, and wind speeds above 1000 cm/sec move sand to new locations. The cycling nature of wind and moving sand determines the shape and location of large sand dunes. At a test site, the prevailing direction of the wind did not change noticeably. However, the velocity did change. Sixty wind speed readings gave an average velocity of 1075cm/sec. Based on long-term experience, σ can be assumed to be 265 cm/sec.

a) Calculate and interpret a 95% confidence interval for the true population mean wind speed at this site.

b) Obtain the interval using Excel and show your output.

c) Does the confidence interval indicate that the population mean wind speed is such that the sand is always moving at this site? Explain.

d) In order to trust the information in the interval, is there anything else about these data that we need to know?

e) What is the margin of error for this interval? Show calculation.

f) If we want to reduce the margin of error to 40 cm/sec, how big must the sample size be?

The coefficient of determination, R2 : Is always negative May be negative or positive Ranges from -1 to +1 Ranges from zero to one Is the ratio of unexplained variation to explained variation Is the ratio of explained variation to unexplained variation has the same sign as the slope of the regression line

In a 1985 study of the relationship between contraceptive use and infertility, 89 of 283 infertile women, compared to 640 of 3833 control (fertile) women, had used an intrauterine device (IUD) at some point in their life. Use the contingency table to test for significant differences in contraceptive use patterns between the two groups. Compute a 95% CI for the difference in the proportion of women who have ever used IUDs between the infertile and fertile women. Compute the OR in favor of ever using an IUD for fertile vs. infertile women. Provide a 95% CI for the true OR corresponding to your answer. What is the relationship between your answers to questions 1 and 4? Need help with question 3 and 4

The scores on a standardized test have an average of 1200 with a standard deviation of 60. A sample of 50 scores is selected.

What is the probability that the sample mean will be between 1195 and 1205? Round your answer to three decimal places.

5. Random pigeonholing

100 pigeons p1,…,p100 fly into 500 labelled holes h1,…,h500. Each pigeon picks a hole uniformly at random and independently from the choices of the other pigeons.

1. What is the probability that at least one hole contains at least 2 pigeons.
   Hint: The answer is approximately 0.9999758457295991
2. What is the probability that at least one hole contains at least 3 pigeons.
   Hint: The answer is approximately 0.4361298523736379.

The fill amount of bottles of a soft drink is normally distributed, with a mean of 2.0 liters and a standard deviation of 0.05 liter. If you select a random sample of 25 bottles, what is the probability that the sample mean will be:

A.) Between 1.99 and 2.0 liters

B.) Below 1.98 liters

C.) Greater than 2.01 liters

D.) The probability is 99% that the sample mean amount of soft drink will be at least how much?

E.) The probability is 99% that the sample mean amount of soft drink will be between which two amounts?

1. According to an educational report, the amount of time that students spend “off-task” (such as by checking their phones) during a one-hour lecture is approximately normally distributed with a mean of 3.2 minutes and a standard deviation of 2.7 minutes. An educator is interested in determining at the α = 0.05 level if the average amount of time his students spend on their phones while he is lecturing differs from the value given in the journal. During a particular class period in which he has 37 students, he noted that the average amount of time students spent on their phones was 4.2 minutes.

(a) State the null and alternative hypotheses for this test.

(b) Compute an appropriate test statistic.

(c) Determine the p-value for this test.

(d) State, in words, your conclusion.

2. Construct a 95% confidence interval for the mean amount of time students spend on their phones. Does this confidence interval support your conclusion from the hypothesis test in part (d)? Why or why not?

Consider a population having a standard deviation equal to 9.94. We wish to estimate the mean of this population.

(a) How large a random sample is needed to construct a 95% confidence interval for the mean of this population with a margin of error equal to 1? (Round your answer to the next whole number.)

The random sample is __________ units.

(b) Suppose that we now take a random sample of the size we have determined in part a. If we obtain a sample mean equal to 345, calculate the 95% confidence interval for the population mean. What is the interval’s margin of error? (Round your answers to the nearest whole number.)

The 95% confidence interval is            [ , ] .

Margin of error = ____________

Playing Roulette In the game of roulette, a wheel consists of 38 slots numbered 0, 00, 1, 2, . . . , 36. To play the game, a ball is spun around the wheel and is allowed to fall into on of the numbered slots. If the number of the slot the ball falls into matches the number you select, you win $35; otherwise you lose $1.

1. Construct a probability distribution for the random variable X, the winnings of each spin.

2. Determine the mean and standard deviation of the random variable X.

3. Suppose that you play the game 100 times, simulate at least 5 possible outcomes. Describe the pattern.

4. Imagine a person who plays the game 1000 times a day, for 365 days. Simulate such a scenario. What is the frequency distribution of that person’s winnings and losings?

5. Suppose that you play the game 1000 times. Describe the sampling distribution of the mean amount won per game.

6. What is the probability of being ahead after playing the game 100 times?

7. What is the probability of being ahead after playing the game 200 times?

8. What is the probability of being ahead after playing the game 1000 times?

9. Based on your investigation, what lesson does this teach you? Write an essay to inform your gambler friend.

Hint: To find the mean and standard deviation:

x <- c(35, -1)
p <- c(1/38, 37/38)

(mu.X <- sum(x*p))

## [1] -0.05263158

(sigma.X <- sqrt(sum((x - mu.X)^2*p)))

## [1] 5.762617

We simulate two possible outcomes if a person plays the game 100 times.

sample(x, 100, replace = TRUE, p)

##   [1] -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
##  [24] -1 -1 -1 -1 35 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
##  [47] -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
##  [70] -1 35 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 35 -1 -1
##  [93] -1 -1 -1 -1 -1 -1 -1 -1

sample(x, 100, replace = TRUE, p)

##   [1] -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
##  [24] -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 35 -1 -1 -1 -1 -1 -1 -1 -1
##  [47] -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
##  [70] -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
##  [93] -1 -1 -1 -1 -1 -1 -1 -1

Repeat this simulation many times and describe the pattern.

Imagine a person who plays the game n=100n=100 times a day, for 365 days. We examine the sum:

day <- 365
daily.sum <- replicate(day,
                       sum(sample(x, 100, replace = TRUE, p)))
(tb <- table(daily.sum))

## daily.sum
## -100  -64  -28    8   44   80  116  152  188 
##   27   63   78   84   62   33   14    2    2

plot(tb)

As the number of games nn increase, the sums follow a normal distribution N(nμ,nσ2)N(nμ,nσ2). With n=1000n=1000 and class width 36,

day <- 365
daily.sum <- replicate(day,
                       sum(sample(x, 1000, replace = TRUE, p)))
(tb <- table(daily.sum))

## daily.sum
## -568 -532 -460 -424 -388 -352 -316 -280 -244 -208 -172 -136 -100  -64  -28 
##    1    1    4    4    4   12   16   16   23   17   27   26   26   24   25 
##    8   44   80  116  152  188  224  260  296  332  368  440 
##   32   19   26   16   12   11    5    4    7    3    1    3

plot(tb/day/36)
curve(dnorm(x, 1000*mu.X, sqrt(1000)*sigma.X), add = TRUE)

The distribution of sample means has a standard deviation of σ/n−−√σ/n:

daily.mean <- daily.sum/1000
hist(daily.mean, probability = TRUE,
     xlim = c(-2, 2), ylim = c(0, 2.5))
curve(dnorm(x, mu.X, sigma.X/sqrt(1000)),
      xlim = c(-2, 2), add = TRUE)

Play 100 times. What is the probability of being ahead?

n1 <- 100
curve(dnorm(x, mu.X, sigma.X/sqrt(n1)), xlim = c(-2, 2),
      ylim = c(0, 2.5), main = "n=100", ylab = "pdf")
abline(v = mu.X)
coord.x <- c(0, seq(0, 2, 0.01), 2)
coord.y <- c(0, dnorm(seq(0, 2, 0.01), mu.X, sigma.X/sqrt(n1)), 0)
polygon(coord.x, coord.y, density = 10)

1 - pnorm(0, mu.X, sigma.X/sqrt(n1))

## [1] 0.4636141

Play 200 times:

n2 <- 200
curve(dnorm(x, mu.X, sigma.X/sqrt(n2)), xlim = c(-2, 2),
      ylim = c(0, 2.5), main = "n=200", ylab = "pdf")
abline(v = mu.X)
coord.x <- c(0, seq(0, 2, 0.01), 2)
coord.y <- c(0, dnorm(seq(0, 2, 0.01), mu.X, sigma.X/sqrt(n2)), 0)
polygon(coord.x, coord.y, density = 10)

1 - pnorm(0, mu.X, sigma.X/sqrt(n2))

## [1] 0.4486139

Play 1000 times:

n3 <- 1000
curve(dnorm(x, mu.X, sigma.X/sqrt(n3)), xlim = c(-2, 2),
      ylim = c(0, 2.5), main = "n=1000", ylab = "pdf")
abline(v = mu.X)
coord.x <- c(0, seq(0, 2, 0.01), 2)
coord.y <- c(0, dnorm(seq(0, 2, 0.01), mu.X, sigma.X/sqrt(n3)), 0)
polygon(coord.x, coord.y, density = 10)

1 - pnorm(0, mu.X, sigma.X/sqrt(n3))

## [1] 0.3863597

A sample of size 18 will be drawn from a population with mean 4 and standard deviation 3.

(a) Is it appropriate to use the normal distribution to find probabilities for x?

(b) If appropriate find the probability that x will be greater than 3.

(c) If appropriate find the 20th percentile of x.

of nine executives in a business firm, four are married, three have never married, and two are divorced. three of the executives are to be selected for promotion. Let Y1 donate the number of married executives and Y2 donate the number of never married executives among the three selected for promotion. Assuming that three are randomly selected from nine available find the joint probability function of Y1 and Y2.

An ecologist hypothesizes that birds with shorter wing spans use wider tree branches. The ecologist captured male birds, measured their wing span and other characteristics in millimeters, and then marked and released them. During the ensuing winter, the ecologist repeatedly observed the marked birds as they foraged for food on tree branches. He noted the branch diameter on each occasion, and calculated the average branch diameter for each bird in centimeters. The measurement data are below. What can the ecologist conclude with an α of 0.01?

a) What is the appropriate statistic?
---Select--- na Correlation Slope Chi-Square
Compute the statistic selected above:

b) Compute the appropriate test statistic(s) to make a decision about H₀.
(Hint: Make sure to write down the null and alternative hypotheses to help solve the problem.)
critical value =  ; test statistic =
Decision:  ---Select--- Reject H0 Fail to reject H0

c) Compute the corresponding effect size(s) and indicate magnitude(s).
If not appropriate, input and/or select "na" below.
effect size =  ;   ---Select--- na trivial effect small effect medium effect large effect

d) Make an interpretation based on the results.

There was a significant positive relationship between the wing span of the birds and branch diameter.There was a significant negative relationship between the wing span of the birds and branch diameter.    There was no significant relationship between the wing span of the birds and branch diameter.

A passcode can have 4 or 5 digits. Digits can be repeated and leading 0s are allowed. So, 1234 would be a 4 digit code that is different from 01234, which is a 5 digit code. How many different passcodes are possible?

A population has a mean of 400 and a standard deviation of 50. Suppose a sample of size 125 is selected and x is used to estimate μ.

a. What is the probability that the sample mean will be within +/- 4 of the population mean (to 4 decimals)?

b. What is the probability that the sample mean will be within +/- 11 of the population mean (to 4 decimals)?

A random sample of 11 items is drawn from a population whose standard deviation is unknown. The sample mean is x¯ = 920 and the sample standard deviation is s = 25. Use Appendix D to find the values of Student’s t.

(a) Construct an interval estimate of μ with 95% confidence. (Round your answers to 3 decimal places.)
  
The 95% confidence interval is from to

(b) Construct an interval estimate of μ with 95% confidence, assuming that s = 50. (Round your answers to 3 decimal places.)
  
The 95% confidence interval is from to

(c) Construct an interval estimate of μ with 95% confidence, assuming that s = 100. (Round your answers to 3 decimal places.)
  
The 95% confidence interval is from to