Stats 1430 Final

Suppose a pizza place claims their pizzas are delivered in 30 minutes and Bob believes it's more than that. Bob samples 49 pizza delivery times at random and gets an average of 35 and sample standard deviation 10 min. What is the value of u0 in this problem? (That's mu sub 0).

30

Stocks have PE ratios that have a normal distribution with mean 15 and standard deviation 5. 10% of the stocks have a PE ratio less than what value?

8.6

Suppose you are doing a hypothesis test Ho: u = 30 vs Ha: u > 30 with n = 10. The sample average is 34 seconds with sample standard deviation of 5.2 seconds and your test statistic is 2.43. Where is your p-value?

Between .025 and .01

If X is a binomial random variable then X is also a _______ random variable

Discrete Finite

Stocks have PE ratios that have a normal distribution with mean 15 and standard deviation 5. What is the probability that a stock has a PE ratio of at most 22?

.9192

Find the mean of X

1.5

Let X be uniform on the interval 0 to 10. What is f(5) and what is P(X=5)?

1/10, 0

Suppose f(x) = kx for X from 0 to 2. Find the value of k

1/2

You want to estimate the average time to do a task for your employees in the mailroom. You take a sample of 10 employees and time them doing this task. The average time is 34 seconds with sample standard deviation of 5.2 seconds. What is the value of t that goes in the confidence interval?

2.2622

Suppose a pizza place claims their pizzas are delivered in 30 minutes and Bob believes it's more than that. Bob samples 49 pizza delivery times at random and gets an average of 35 and sample standard deviation 10 min. What is the test statistic?

3.5

You want to estimate the average amount of revenue from all the bank branches of your company in the U.S. You want the margin of error to be no more than 1000. Assume the standard deviation is 30,000. How many branches should be sampled?

3458

Suppose X has a normal distribution with mean 80 and standard deviation 5. Then about 95% of the values must lie between what two numbers?

70 and 90 By the 68-95-99.7 rule, if X has a normal distribution then about 95% of the values lie within 2 standard deviations of the mean. That means 2 x 5 = 10 on each side of the mean, or between 80-10 = 70 and 80+10 = 90.

Exam scores have a mean of 70 and standard deviation of 10. The professor wants to give everyone a 10% curve added to their score. What is the new mean?

77 Y=1.10x Mean of Y=Mean of 1.10X=1.0 Mean X = 1.10(70)= 77

Suppose X is binomial with n = 5 and p = .3. Find the probability that X is at most 4.

.9976

Suppose X is uniform on the interval 0 to 20. What is f(x)?

1/20 If f(x) is uniform that means it is a rectangle whose length goes from 0 to 20. To make the total area 1, the height has to be 1/20. That's f(x).

Bob guesses randomly on a 20 question multiple choice test where there are 4 possible answers for each question. This is a binomial distribution where Yes = got it right, and No = got it wrong. What is the value of p?

1/4

Suppose you want to estimate the average price of gas in Ohio and your 95% confidence interval was (1.20, 1.50). How confident are you that the SAMPLE MEAN is in this interval?

100%

You want to estimate the average time to do a task for your employees in the mailroom. You take a sample of 10 employees and time them doing this task. The average time is 34 seconds with sample standard deviation of 5.2 seconds. What is the margin of error?

3.72

Bob guesses randomly on a 20 question multiple choice test where there are 4 possible answers for each question. What is the variance of the number of answers Bob gets right?

3.75

On a regression line, you can find a residual for any value anywhere on the entire line. True or false?

False You can only find a residual where there is a data point. You need observed - predicted for a residual. Predicted is the value on the line, and observed is the value of the data point.

What happens to the standard error of X-bar as n increases to infinity?

It goes to 0

A confidence interval gets wider when you don't know the population standard deviation because you have to use t instead of Z.

True You have to pay a penalty for not having the population standard deviation; the penalty is a value that is greater than 1.96 for 95% confidence intervals. It comes from the t-distribution with n-1 degrees of freedom.

last year the average time to do a task for your employees was 30 seconds, and you are afraid it's now more than that. You take a sample of 10 employees and time them doing this task. The average time is 34 seconds with sample standard deviation of 5.2 seconds. True or false: Your test statistic is 2.43.

True t10-1 = t9 = (34-30)/(5.2/sqrt10)=2.43

Let X be a continuous random variable with f(x) = x/2 on 0 to 2. True or false? The median of X is the square root of 2.

True We know the integral from 0 to the median equals .50 because half of the area is below the median. Let the median = M. Set up the integral and solve for M. integral (x/2) from 0 to M = .50 x^2/(2x2) evaluated at 0 and M = .50 x^2/4 from 0 to M = .50 M^2/4 - 0 = .50 M^2 = 2 M = sqrt (2)

Which of the following requires independence?

Var (X-Y) = Var(X) + Var (Y) If X and Y are dependent then Var (X-Y) is a longer formula involving rho (Greek small r), whose last term does not drop out (see formula sheet)

X is a normal random variable with mean 10 and standard deviation 3 Y is a normal random variable with mean 20 and standard deviation 4 X and Y are independent. Find the mean and standard deviation of X - Y.

-10 and 5

Suppose a pizza place claims their pizzas are delivered in 30 minutes and Bob believes it's more than that. Bob samples 49 pizza delivery times at random and gets an average of 35 and sample standard deviation 10 min. What are the hypotheses in this problem?

Ho: u=30 and Ho: u>30

Let X have a normal distribution with mean 70 and standard deviation 5. Let X-bar be the mean of 20 values chosen at random from it. What is bigger, P(X > 75) or P(X-bar>75)?

Prob for X is larger

When you take all possible sample means from all possible samples of size n from a population and put them all together, what do you have?

Sampling distribution

What element of X-bar is addressed by the Central limit theorem, the shape, mean, or standard deviation?

Shape only

Let X be a continuous random variable from 0 to 2 with f(x)= x/2. What is the mean of x?

Some number between 1 and 2

If you are counting the number of RED M&Ms in a bag you have a binomial distribution.

True

Suppose Bob questions whether cereal boxes are being filled correctly, and he believes they are being underfilled. He collects his data, finds his test statistic and p-value, and ends up failing to reject Ho. He didn't have enough evidence to say the cereal boxes were being underfilled. What Type of error could Bob have made?

Type 2 When you fail to reject Ho, there is a chance you are wrong. That chance is called a Type 2 error. With a Type 2 error you have a missed detection - Ho was really false but you didn't reject it. Bob didn't have enough evidence that the cereal boxes were underfilled, although they really were.

True or False? Z and t can both take on negative numbers.

Z and t can both be negative

What conditions need to be checked before you use Z to solve a binomial problem with a large n?

np and n(1-p) at least 10

Which distribution has a larger standard deviation, the Z or the t?

t

Suppose Bob conducted a hypothesis test for the mean where Ho: mu = 20 and Ha: mu < 20, and found a test statistic of -1.28. What is his p-value?

.1003 Probability of being "beyond" (in this case less than) the test statistic is P(Z < -1.28) = .5 - .3997 = .1003.

Research says that 80% of companies start making a profit after 3 years. You take a random sample of 100 companies. what's the chance that at most 75 of them make a profit after 3 years?

.1056

What is the probability that Z is at least 1.25?

.1056

Suppose you want to estimate the average price of gas in Ohio and your 95% confidence interval was (1.20, 1.50). What is the margin of error for this confidence interval?

.15

Suppose 90% of students are at home over the summer. You randomly select 2 students. What's the chance that exactly one of them is at home over the summer?

.18

Suppose Bob conducted a hypothesis test for the mean where Ho: mu = 20 and Ha: mu not equal to 20, and found a test statistic of -1.28. What is his p-value?

.2006 Double the probability of being beyond the test statistic since Ha is not equal to. Prob of being beyond -1.28 is P(Z < -1.28) = .5 - .3997 = .1003. Now double it to get .2006.

Suppose Bob conducted a hypothesis test for the mean where Ho: u = 80 vs Ha: u > 80 and he gets a p-value of .06. What does Bob decide to do about Ho?

Fail to reject Ho Since p-value = .06 > alpha = .05, we fail to reject Ho. Not enough evidence to say Ha is true.

If you did a t-test and got a p-value between .03 and .07, what do you conclude?

Fail to reject Ho Since the p-value is between .03 and .07, the ENTIRE range is not less than alpha = .05, so you fail to reject Ho. You need the ENTIRE p-vaue range to fall below .05 to reject Ho.

If you count the number of M&Ms of each color you have a binomial distribution.

False

P(Z<-1) = 1 - P(Z>1)

False

Suppose X is a continuous random variable. Then f(x) must be between 0 and 1 for all values of X.

False

The NORMAL PROBABILITY PLOT should look like a bell-shaped curve if the data have a normal distribution.

False

Let X be a continuous random variable with f(x) = x/2 for X and 0 to b. Find the value of b

b=2 For f(x) to be legitimate, its integral from 0 to b must equal 1. Set this up and solve for b. Integral(x/2) = x^2/(2x2) = x^2/4 evaluated from 0 to b. So b^2/4 - 0 = 1 So b^2 = 4 and b = 2. f(x) = x/2 from 0 to 2.

The t-distribution looks more and more like the Z distribution as n gets larger and larger.

true