**# 1**

The web publisher **california-wine-regions.com**provides information on travelling to wine regions in the state of California. Access to the website is free but revenues are generated by selling ads that are posted on the website. In the following month, the website has committed to display ads to 650,000 viewers, i.e., 650,000 impressions. Based on data from previous months the traffic to the website is estimated to be normally distributed with a mean of 850,000 viewers and a standard deviation of 150,000.

A. What is the probability that the web publisher will be able to deliver the promised impressions?

B. How many impressions should the web publisher have taken on, to be able to guarantee a 95% service level?

**# 2**

In a large class in statistics, the final examination grades have a mean of 67.4 and a standard deviation of 12. Assuming that the distribution of these grades is normal, find:

A. the percentage of grades that should exceed 85

B. the percentage less than 45

C. the number of passes (pass mark is 50) in a class of 180

D. the lowest distinction mark if the highest 8% of grades are to be regarded as distinctions

**# 3**

A sample of 36 weekly observations of the FTSE 100 index returns has a mean of 0.005 (0.5%) and a standard deviation of 0.02 (2%).

A. Calculate a 95% confidence interval for the mean weekly return.

B. How large a sample is required to estimate the mean weekly return to within a maximum error bound of ±0.004 (0.4%)?

C. Do we need to assume that the weekly returns follow a normal distribution?

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**# 4**

An experiment involves selecting a random sample of 256 middle managers for study. One item of interest is annual income. The sample mean is computed to be $35,420, and the sample standard deviation is $2,050.

A. What is the estimated mean income of all middle managers (the population)?

B. Give a 95 percent confidence interval (rounded to the nearest $10) for your estimate of the mean income. Do you have to make any assumptions?

C. Interpret the meaning of the confidence interval.

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**# 5**

Please discuss and write down the answers to the following questions:

A. If you collect 4 times more data, how much narrower will your confidence interval (CI) be? Same question for collecting 100 times more data.

B. Assume you work for a manager who says one day “I got the budget to collect twice as much data; that’s great because our estimates will be twice as precise.” Is anything wrong with his statement?

C. Your manager says “Let’s just calculate our CIs with 90% coverage probability instead of 95%; this will make the CIs narrower.” Is she right or wrong? Your manager adds: “We get better precision this way.” What is the manager’s misconception?