A number between 1 and 10, inclusive, is randomly chosen. Events A and B are defined as follows. A: {The number is even} B: {The number is less than 7} Identify the simple events comprising the event (A and B). Select one: {1, 2, 3, 4, 5, 6} {2, 4, 6, 8, 10} IncorrectIncorrect {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} {2, 4, 6} Question 10 Incorrect 0.00 points out of 1.00 Not flaggedFlag question Question text A number between 1 and 10, inclusive, is randomly chosen. Events A and B are defined as follows. A: {The number is even} B: {The number is less than 7} Identify the simple events comprising the event (A or B). Select one: {1, 2, 3, 4, 5, 6, 8, 10} {1, 2, 3, 4, 5, 6, 7, 8, 10} {1, 2, 3, 4, 5, 6} IncorrectIncorrect {2, 4, 6}

In the following problem, check that it is appropriate to use the normal approximation to the binomial. Then use the normal distribution to estimate the requested probabilities.

Do you take the free samples offered in supermarkets? About 64% of all customers will take free samples. Furthermore, of those who take the free samples, about 41% will buy what they have sampled. Suppose you set up a counter in a supermarket offering free samples of a new product. The day you were offering free samples, 309 customers passed by your counter. (Round your answers to four decimal places.)

(a) What is the probability that more than 180 will take your free sample?


(b) What is the probability that fewer than 200 will take your free sample?


(c) What is the probability that a customer will take a free sample and buy the product? Hint: Use the multiplication rule for dependent events. Notice that we are given the conditional probability P(buy|sample) = 0.41, while P(sample) = 0.64.


(d) What is the probability that between 60 and 80 customers will take the free sample and buy the product? Hint: Use the probability of success calculated in part (c).

Prob 1Coffee Beanery in Chicago  is open 365 days a year and sells an average of 120 pounds of Kona Coffee beans a day (Assume demand is to normally distributed with a standard deviation of 40 pounds/day). Beans are ordered from Hawaii, and will arrive in exactly 4 days with a flat rate shipping and handling charge of $150. Per-pound annual holding costs for the beans are $3. Coffee is highly profitable, so the Beanery would like to have at most a 1% chance of running out of beans. Orders should be placed in whole pounds, but otherwise the Kona suppliers can deal with orders of any size. .

1)Calculate the economic order quantity (EOQ) for Kona coffee beans ________lbs

2)Calculate the total annual holding costs of cycle stock for Kona coffee beans $__________

3) Calculate the total annual fixed ordering costs for Kona coffee beans $__________

4)Calculate the Re-Order Point ________lbs

5)If management specified that a 2% stock-out risk is now okay, would safety stock holding costs decrease, increase, remain unchanged or do we not have enough info to tell? (Circle 1)

Problem 1-contd' Now assume that Kona supplier wants to place the Beanery on a fixed order intervals, where they will only deal with orders every 3 weeks (Hawaiians have better things to do than be waiting by the phone for obnoxious Manhattanites). Assume all other parameters remain the same.

1)If the Beanery has only 350lbs of beans left on the day they are allowed to place an order, calculate the probability they will run out before this next order arrives. ________%

2)Compare the cost of safety stock for the Fixed Order interval to that associated with the reorder point. How does it differ and why?

A sample of final exam scores is normally distributed with a mean equal to 20 and a variance equal to 25. a) what percentage of scores is between 15 and 25. b) what raw score is the cutoff for the top 10% of scores. c) what is the probability of a score less than 27. d) what is the proportion below 13.

random election of 11 children tested and finds that their mean attention span is 31 minutes with a standard deviation of 8 minutes. assuming attention spans are normally distributed, find a 95% confidence interval for the mean attention span of children.   also calculate the upper and lower limit of the confidence interval

A random sample of n = 1,400 observations from a binomial population produced x = 659.

(a) If your research hypothesis is that p differs from 0.5, what hypotheses should you test?

H₀: p ≠ 0.5 versus H_a: p = 0.5H₀: p = 0.5 versus H_a: p < 0.5     H₀: p = 0.5 versus H_a: p > 0.5H₀: p < 0.5 versus H_a: p > 0.5H₀: p = 0.5 versus H_a: p ≠ 0.5

(b) Calculate the test statistic and its p-value. (Round your test statistic to two decimal places and your p-value to four decimal places.)

Child Health and Development Studies (CHDS) has been collecting data about expectant mothers in Oakland, CA since 1959. One of the measurements taken by CHDS is the age of first time expectant mothers. Suppose that CHDS finds the average age for a first time mother is 26 years old. Suppose also that, in 2015, a random sample of 50 expectant mothers have mean age of 26.5 years old, with a standard deviation of 1.9 years. At the 5% significance level, we conduct a one-sided T-test to see if the mean age in 2015 is significantly greater than 26 years old. Statistical software tells us that the p-value = 0.034.

Which of the following is the most appropriate conclusion?

A) There is a 3.4% chance that a random sample of 50 expectant mothers will have a mean age of 26.5 years old or greater if the mean age for a first time mother is 26 years old. B) There is a 3.4% chance that mean age for all expectant mothers is 26 years old in 2015.

C) There is a 3.4% chance that mean age for all expectant mothers is 26.5 years old in 2015.

D) There is 3.4% chance that the population of expectant mothers will have a mean age of 26.5 years old or greater in 2015 if the mean age for all expectant mothers was 26 years old in 1959.

using chebyshev's inequality, the probability that a random variable will be within 2 standard deviations of its own mean the least?

The distribution of scores on a recent test closely followed a Normal Distribution with a mean of 22 points and a standard deviation of 2 points. For this question, DO NOT apply the standard deviation rule.

(a) What proportion of the students scored at least 25 points on this test, rounded to five decimal places?

(b) What is the 20 percentile of the distribution of test scores, rounded to three decimal places?

A consumer product testing organization uses a survey of readers to obtain customer satisfaction ratings for the nation's largest supermarkets. Each survey respondent is asked to rate a specified supermarket based on a variety of factors such as: quality of products, selection, value, checkout efficiency, service, and store layout. An overall satisfaction score summarizes the rating for each respondent with 100 meaning the respondent is completely satisfied in terms of all factors. Suppose sample data representative of independent samples of two supermarkets' customers are shown below.

(a)

Formulate the null and alternative hypotheses to test whether there is a difference between the population mean customer satisfaction scores for the two retailers. (Let μ₁ = the population mean satisfaction score for Supermarket 1's customers, and let μ₂ = the population mean satisfaction score for Supermarket 2's customers. Enter != for ≠ as needed.)

H₀:

H_a:

(b)

Assume that experience with the satisfaction rating scale indicates that a population standard deviation of 14 is a reasonable assumption for both retailers. Conduct the hypothesis test.

Calculate the test statistic. (Use μ₁ − μ₂. Round your answer to two decimal places.)

Report the p-value. (Round your answer to four decimal places.)

p-value =

At a 0.05 level of significance what is your conclusion?

Reject H₀. There is not sufficient evidence to conclude that the population mean satisfaction scores differ for the two retailers.Do not reject H₀. There is sufficient evidence to conclude that the population mean satisfaction scores differ for the two retailers.     Reject H₀. There is sufficient evidence to conclude that the population mean satisfaction scores differ for the two retailers.Do not reject H₀. There is not sufficient evidence to conclude that the population mean satisfaction scores differ for the two retailers.

(c)

Which retailer, if either, appears to have the greater customer satisfaction?

Supermarket 1 Supermarket 2     neither

Provide a 95% confidence interval for the difference between the population mean customer satisfaction scores for the two retailers. (Use x₁ − x₂.Round your answers to two decimal places.)

_______ to ________

use R

# Problem 4 (5 pts each):
# Set x as a vector of 500 random numbers from Unif(100,300).
# This vector will be kept fixed for the rest of this problem.
#
# (a) Define a function b1(x, beta0, beta1, sigm) that uses the lm() function to
# return the regression line slope b1 for y as a linear function of x, where
#
# y = beta0 + beta1 x + err
#
# and the error term 'err' has a normal N(0,sigm^2) distribution
# (note that standard deviation is equal 'sigm').
#
# Hint: See how the slope b1 is extracted in the initial example of Session 11.

# (b) Replicate the function b1 twenty thousand times for
# beta0 = 15, beta1 = 2, and sigm =10, and store into a vector 'Slopes'.

# (c) Plot the empirical density of Slopes.

# (d) Calculate sample mean and sample variance of Slopes.

# (e) Add to the plot the pdf of a Normal distribution with parameters from part (d).

Many hotels have begun a conservation program that encourages guests to re-use towels rather than have them washed on a daily basis. A recent study examined whether one method of encouragement might work better than another. Different signs explaining the conservation program were placed in the bathrooms of the hotel rooms, with random assignment determining which rooms received which sign. One sign mentioned the importance of environmental protection, whereas another sign claimed that 75% of the hotel’s guests choose to participate in the program. The researchers suspected that the latter sign, by appealing to a social norm, would produce a higher proportion of hotel guests who agree to re-use their towels. Researchers used the hotel staff (a mid-sized, mid-priced hotel in the Southwest that was part of a well-known national hotel chain) to record whether guests staying for multiple nights agreed to reuse their towel after the first night.

(a) Identify the observational units, explanatory variable, and response variable in this study.

(b) State the null and alternative hypotheses in symbols, and be sure to define the parameter in the context of this study.

The following table displays the observed data in this study:

(c) Calculate the conditional proportions of re-use in each group.

(e) Use a two-sample z-test to test the hypotheses that you stated in (a). Report the test statistic and p-

value.

(f) Report your test decision at the α = 0.10, 0.05, and 0.01 significance levels. Also summarize what

these test decisions reveal about the strength of evidence for the researchers’ conjecture.

(g) Produce and interpret a 90% confidence interval for the difference in probabilities of re-using towels

between these two signs.

PROBLEM 2.

Based on data from Consumer Reports, replacement times of TVs is on average 3.4 years with the standard deviation of 1.2 years. Answer the following questions.

Question 2 (2 points):

A random sample of 41 TVs is selected. Find the probability that their average replacement time is between 3 and 4 years.

a) 0.6472
b) 0.9827
c) 0.3208
d) None of the above

Question 3 (2 points):

If a random sample of 29 TVs was chosen, what would be the probability that their average replacement time exceeds 3.5 years?

a) 0.6736
b) 0.3264
c) 0.4681
d) None of the above

Listed below are attractiveness ratings made by participants in a speed dating session. Each attribute rating is the sum of the ratings of five attributes (sincerity, intelligence, fun, ambition, shared interests).

Use a 0.05 significance level to test the claim that there is a difference between female attractiveness ratings and male attractiveness rating by following the steps below:

(a) State the null and alternative hypotheses, indicate the significance level and the type of test (left-, right-, or two-tailed test).

(b) Calculate by hand the test statistic.

(c) Use the appropriate sheet in the Hypothesis Test and Confidence Interval template to complete all relevant computations (including the test statistic: compare with (b) to confirm your calculation is correct).

(d) Use the P-value obtained in (c) to explain whether or not the null hypothesis is rejected.

(e) What can be concluded based off this data?

(f) Are there any potential issues related to the validation of the result (Hint: the subjective nature of the measures)

Instructions This assignment is to be typed up in the supplied R-Script. You need to show all of your work in R in the given script.

3. Infant mortality. The infant mortality rate is defined as the number of infant deaths per 1,000 live births. This rate is often used as an indicator of the level of health in a country. The relative frequency histogram below shows the distribution of estimated infant death rates for 224 countries for which such data were available in 2014.

(a) Estimate Q1, the median, and Q3 from the histogram.

(b) Would you expect the mean of this data set to be smaller or larger than the median? Explain your reasoning.

(c) If you calculated the z-score for the median in this distribution, would the result be positive or negative? Explain your reasoning.