3) When I lived in California I had a small lemon tree in the front yard. If we had rain in the summer (rare=P=.2) it would yield up to 10 lemons, distributed with a binomial distribution, N= 10, P=0.6. If it is perfectly dry (most of the time P=0.8) it the distribution would be binomial with N=6, P=0.4 a) If you get 3 lemons what the probability that it rained. b) If you have 4 lemons, what is the probability that it rained.

2. A firm is considering the delivery times of two raw material suppliers, A and B. The firm is basically satisfied with supplier A; however, if the firm finds the mean delivery time of supplier B is less than the mean delivery time of supplier A, the firm will begin purchasing raw materials from supplier B (meaning, switch from supplier A to supplier B). Independent samples (assume equal population variances) show the following sample data for the delivery times of the two suppliers:

a. State the null and alternative hypotheses for this situation.
b. Describe what a Type I Error would be in this situation (please be as specific as possible).
c. If α = 0.01, what is the critical value of the associated test statistic?
d. What is the calculated value of the associated test statistic?
e. State your decision about the null hypothesis by comparing the critical and calculated values of the test statistic (Parts c and d).
f. What action do you recommend in terms of supplier selection?

4. For this problem, you’ll compare the hypergeometric and binomial distributions. Suppose there is a sock drawer with N socks, each placed loosely in the drawer (not rolled into pairs). The total number of black socks is m. You take out a random sample of n < m socks. Assume all the socks are the same shape, size, etc. and that each sock is equally likely to be chosen.

(a) Suppose the sampling is done without replacement. Calculate the probability of getting at least 2 black socks (your goal in order to wear matching black socks that day...) under the following conditions:

(i) N = 10, n = 4, m = 5.

(ii) N = 20, n = 4, m = 10.

(iii) N = 40, n = 4, m = 20.

(b) Suppose the sampling is done with replacement (this doesn’t make much sense if you are planning to wear the socks!). Calculate the probability of getting at least two black socks when you sample four socks and the proportion of black socks is 0.5. Compare your answer to those in (a).

7. The mean weekly earnings for employees in general automotive repair shops is $450 and the standard deviation is $50. A sample of 100 automotive repair employees is selected at random.
a. Find the probability that the mean earnings is less than $445.

b. Find the probability that the mean earning is between $445 and $455.

c.Find the probability that the mean earnings is greater than $460.

8. A drug manufacturer states that only 5% of the patients using a high blood pressure drug will experience side effects. Doctors at a large university hospital use the drug in treating 200 patients.
a.What is the probability that 15 or fewer patients will experience a side effect?

b. What is the probability that between 7 and 12 patients will experience a side effect?

1. Obtain a linear regression equation for the data to predict the mean temperature values for any given CO2 level. How good is the linear fit for this data? Explain using residual plot and R-square value. To draw residual plot, compute the estimated temperatures for every value of the CO2 level using the regression equation. Then compute the difference between observed (y) and estimated temperature values (called residual; ). Plot the residuals versus CO2 level (called a residual plot).

​a) If the confidence interval for the difference in population proportions p1​ - p2 includes​ 0, what does this​ imply? ​

b) If all the values of a confidence interval for two population proportions​ are​ positive, then what does​ this​ imply? ​

c) If all the values of a confidence interval for two population proportions​ are​ negative, then what does​ this​ imply?

d) Explain the difference between sampling with replacement and sampling without replacement. Suppose you had the names of 10​ students, each written on a 3 by 5​ notecard, and want to select two names. Describe both procedures.

Problem 12-09

The Iowa Energy are scheduled to play against the Maine Red Claws in an upcoming game in the National Basketball Association Developmental League (NBA-DL). Because a player in the NBA-DL is still developing his skills, the number of points he scores in a game can vary dramatically. Assume that each player's point production can be represented as an integer uniform variable with the ranges provided in the table below.

1. Develop a spreadsheet model that simulates the points scored by each team. What is the average and standard deviation of points scored by the Iowa Energy? If required, round your answer to one decimal place.
   
   Average =
   
   Standard Deviation =
   
   What is the shape of the distribution of points scored by the Iowa Energy?
   
   Bell Shaped
   
2. What is the average and standard deviation of points scored by the Maine Red Claws? If required, round your answer to one decimal place.
   
   Average =
   
   Standard Deviation =
   
   What is the shape of the distribution of points scored by the Maine Red Claws?
   
   Bell Shaped
   
3. Let Point Differential = Iowa Energy points - Maine Red Claw points. What is the average point differential between the Iowa Energy and Maine Red Claws? If required, round your answer to one decimal place. Enter minus sign for negative values.
   
   
   
   What is the standard deviation in the point differential? Round your answer to one decimal place.
   
   
   
   What is the shape of the point differential distribution?
   
   Bell Shaped
   
4. What is the probability of that the Iowa Energy scores more points than the Maine Red Claws? If required, round your answer to three decimal places.
   
   
   
5. The coach of the Iowa Energy feels that they are the underdog and is considering a "riskier" game strategy. The effect of the riskier game strategy is that the range of each Energy player's point production increases symmetrically so that the new range is [0, original upper bound + original lower bound]. For example, Energy player 1's range with the risky strategy is [0, 25]. How does the new strategy affect the average and standard deviation of the Energy point total? Round your answer to one decimal place.
   
   Average =
   
   Standard Deviation =
   
   Explain.
   
   The input in the box below will not be graded, but may be reviewed and considered by your instructor.
   
   
   
   How is the probability of the Iowa Energy scoring more points that the Maine Red Claws affected? If required, round your answer to three decimal places.
   
   Probability =

. The learning styles of students in a university biology course were measured and the students were divided into two groups depending on their propensity towards visual learning (style 1) versustext-based learning (style 2). Severalteaching interventions aimed at visual learners were introduced into the course and the instructors scored the students as to whether or not they showed learning gains compared to the usual text-based materials. The results are given in the table below. Conduct a Chi-square analysis to determine if learning gains were contingent of learning style. Discuss the results and state your conclusions. LS1/Yes:28, LS1/No:10, LS2/Yes:48, LS2/No:114

According to the Kentucky Transportation Cabinet, an average of 167,000 vehicles per day crossed the Brent Spence Bridge into Ohio in 2009. Give the state of disrepair the bridge is currently under, a journalist would like to know if the mean traffic count has increased over the past five years. Assume the population of all traffic counts is bimodal with a standard deviation of 15,691 vehicles per day.
a. What conjecture would the journalist like to find support for in this sample of vehicles?
b. A sample of 75 days is taken and the traffic counts are recorded, completely describe the sampling distribution of the sample mean number of vehicles crossing the Brent Spence Bridge. Type out all supporting work.
c. The sample of 75 days had an average of 172,095.937 vehicles crossing the bridge. What is the probability of observing a sample mean of 172,095.937 vehicles or larger? Type out all supporting work.
d. Based on the probability computed in part c, what can be conclusion can be made about the conjecture? Explain.
e. If the number of sampled days was changed to 25, how would the shape, mean, and standard deviation of the sampling distribution of the sample mean traffic counts be affected?

We have five groups and three observations per group. The group means are 6.5, 4.5, 5.7, 5.7, and 5.1, and the mean square for error is .75. Compute an ANOVA table for these data.

The method of tree ring dating gave the following years A.D. for an archaeological excavation site. Assume that the population of x values has an approximately normal distribution.

(a) Use a calculator with mean and standard deviation keys to find the sample mean year x and sample standard deviation s. (Round your answers to the nearest whole number.)

(b) Find a 90% confidence interval for the mean of all tree ring dates from this archaeological site. (Round your answers to the nearest whole number.)

(a) Write an R function rnormmax that has three arguments, n, mu, and sigm,
# and returns the maximum of a vector of n random numbers from the normal distribution
# with mean mu and standard deviation sigm. Make the arguments mu and sigm optional
# with default values of 0 and 1, respectively.

# (b) Write an R code that replicates rnormmax(n=1000) hundred thousand times and creates
# a histogram of the resulting vector via standard hist function.

The Downtown Parking Authority of Tampa, Florida, reported the following information for a sample of 229 customers on the number of hours cars are parked and the amount they are charged.

a-1. Convert the information on the number of hours parked to a probability distribution. (Round your answers to 3 decimal places.)

a-2. Is this a discrete or a continuous probability distribution? Discrete Continuous

b-1. Find the mean and the standard deviation of the number of hours parked. (Do not round the intermediate calculations. Round your final answers to 3 decimal places.)

b-2. How long is a typical customer parked? (Do not round the intermediate calculations. Round your final answer to 3 decimal places.)

Find the mean and the standard deviation of the amount charged. (Do not round the intermediate calculations. Round your final answers to 3 decimal places.)

Conditions for instruments to be valid include all of the following except: (A) each one of the instrumental variables must be normally distributed. (B) at least one of the instruments must enter the population regression of ?? on the ??s and the ??s. (C) perfect multicollinearity between the predicted endogenous variables and the exogenous variables must be ruled out. (D) each instrument must be uncorrelated with the error term

1. Materials and Introduction:
   1. Each person should have 10 KISSES^® chocolates of the same variety and a 16-ounce plastic cup
   2. Examine one of the KISSES^® chocolates. There are two possible outcomes when a KISSES^® chocolate is tossed - landing completely on the base or not landing completely on the base.
   3. Estimate p, the proportion of the time that a KISSES^® chocolate will land completely on its base when tossed.
   4. We will assume that p is approximately 50% and test the claim that the population proportion of Kisses® chocolates that land completely on the base is less than 50%.
   5. We will assume that p is approximately 35% and test the claim that the population proportion of Kisses® chocolates that land completely on the base is different than 35%.

2. Experiment: The investigation is as follows:

6. Put 10 KISSES^® chocolates into the cup
7. Gently shake the cup twice to help mix up the candies.
8. Tip the cup so the bottom of the rim is approximately 1 – 2 inches from the table and spill the candies.
9. Count the number of candies that land completely on their base.
10. Return the candies to the cup and repeat until you have spilled the candies 5 times.
11. Record your results on the Data Table.

Data Table:

3. Questions

We treat the 50 results for each student as 50 independent trials. Actually, each student has ten independent trials of 5 tosses each. We make the assumption that the 10 tosses within a trial are roughly independent to expedite data collection.

1. We will assume that p is approximately 50% for the following two tests.
   1. Test the claim that the population proportion of Kisses® chocolates that land completely on the base is different than 50% at α = 10% level of significance.

State hypotheses and α:

Calculate the evidence – State test used. Clearly state the p-value.

State the complete decision rule then state clearly your decision.

State your conclusion in context to the problem.

2. Test the claim that the population proportion of Kisses® chocolates that land completely on the base is less than 50% at α = 10% level of significance.

State hypotheses and α:

Calculate the evidence – State test used. Clearly state the p-value.

State the complete decision rule then state clearly your decision.

State your conclusion in context to the problem.

2. We will assume that p is approximately 35% for the following two tests.

1. Test the claim that the population proportion of Kisses® chocolates that land completely on the base is different than 35% at α = 5% level of significance.

State hypotheses and α:

Calculate the evidence – State test used. Clearly state the p-value.

State the complete decision rule then state clearly your decision.

State your conclusion in context to the problem.