Question:

A random sample of 40 Americans yielded a sample mean of 67 gallons of carbonated beverages per year. Assume that the population standard deviation is 22 gallons per year, and the consumption of carbonated beverages per year follows a normal distribution. List and check the conditions, calculate a 95% confidence interval, and write it in a complete sentence related to the scenario.

The University of Greatlake Surveys of Consumers found that the mean index of consumer sentiment for December 2015 was 82.9. Suppose the population is normal and this result was based on a random sample of size 85, which had a standard deviation of 12. List and check the conditions, construct a 95% confidence interval for the population mean consumer sentiment, and write it in a complete sentence related to the scenario.

A survey of customers at a local supermarket asks whether they found shopping at this market more pleasing than at a nearby store. Of the 3000 forms distributed to customers, 429 were filled in and 276 of these said that the experience was more pleasing. With 90% confidence, list and check the conditions, calculate the confidence interval to estimate the true proportion of customers that consider this market more pleasing than the other store, and write it in a complete sentence related to the scenario.

The service department at a new car dealer checks for small dents in cars brought in for scheduled maintenance. It finds that 27 of 86 randomly-sampled cars have a dent that can be removed easily. The service department wants to estimate the percentage of all cars with these easily repaired dents. List and check the conditions, construct a plus-four confidence interval to estimate the percentage with 99% confidence, and write it in a complete sentence related to the scenario.

A political candidate is anxious about the outcome of the next election. To have his next survey result produce a 95% confidence interval with margin of error of no more than 0.03, how many eligible voters are needed in the sample?

A. In social science, a Pearson coefficient of .75 means that the two variables are not related. True or false ? B. R^2 in the bivariate model is the percent of the the variance in Y explained by X. True or False? C. When you standardize both variables in a bivariate regression, the intercept will always be zero. True or false? D. In multivariate regression, there is only one e term. True or false?Q2. My home uses two light bulbs. On average, a light bulb last for 22 days (expo- nentially distributed). When a light bulb burns out, it takes and average of 2 days (exponentially distributed] before I replace the bulb. (a) Formulate a three-state birth and death process of this situation. (b Determine the fraction of the trne that both light bulbs are working. ) ((3) Determine the fraction of the time that no light bulbs are working. ) ((1 Suppose that Whenever a bulb burns out, a fixed failure cost of 5 dollars is incurred. Moreover, when any of the bulb is not working, an inconvenience cost of 1 dollar per bulb per day is incurred. Determine the long run cost per day for this system. 3. (a) During lunch hour, arrivals of customers at a pizza hut restaurant follows a Poisson process with the rate of 12D customers per hour. The restaurant has one line, with three workers taking food orders at independent services stations. Each worker takes an exponen- tialll'rr distributed amount of time-on average 1 minute-to serve a customer. Let X: denote the number of customers in the restaurant {in line and being serviced) at time t. Then, the process (X: : t 2; ] is a continuous-time Markov chain. (i) Show that the process is a birth-anddeath process by giving the birth and death rates. [4 marks] (ii) Find the generator matrix of the above process. [5 marks] (iii) For each integer k 3: I}, derive the long-term probability that there are It customers in the restaurant. [3 marks] [iv] Calculate the long-term probabilityr that all three workers are busy. [6 marks] (v) Find the average number of customers in the restaurant in the long term. [1" marks]