Question 1

According to one study, it has been claimed that 70% of all single men in Montreal would welcome a woman taking the initiative in asking for a date. You decide to challenge this study by performing a little experiment of your own. You randomly select 15 single men from around Montreal and ask them if they would be comfortable with a woman asking them out for a date. If the claim made in the study was true, what is the probability that among the sample:

a) Exactly 10 men would say yes?
b) Exactly 5 men would say no?
c) More than 11 men would say yes?
d) At most 4 men would say no?
e) Between 6 and 8 men would say yes?

Question 2

Over the course of a year, the manager of Motel St-Jacques, located in Brossard, Quebec, has noticed that on average, 180 rooms will be rented out every night, with a standard deviation of 15 rooms. If it is assumed that the probability distribution for the number of room rentals forms a normal distribution, what is the probability that on a given night:

a) Less than 150 rooms are rented.
b) More than 195 rooms are rented.
c) More than 164 rooms are rented.
d) Between 170 and 200 rooms are rented.

Question 3

You have been studying the effects of different fertilizers on the growth of certain produce. In the latest instalment of your experiment, you are testing Quebec's own "Gro-Haut" fertilizer and its effects on carrots. From the sample of carrots that you have measured, you calculate an average length of 28 cm with a standard deviation of 6. Please construct a confidence interval for the mean length of the carrots using Gro-Haut in the following circumstances:

a) Sample size = 35, confidence level = 98%.
b) Sample size = 15, confidence level = 90%.
c) What would be the minimum sample needed to estimate the population mean of the length of
carrots using Gro-Haut to within 1.5 cm with 99% confidence?

Question 4

An automatic puck shooting machine is a device that fires hockey pucks at specified speeds. The machine's manufacturer, Puck Chuck, constructed the device so that there was little variance in the velocities of the projectiles. For example, if a machine was to be set at a mean output of 90 miles per hour (mph), its standard deviation would remain 4 mph. Assuming that actual speeds of the device are normally distributed, please answer the following questions about a machine set to shoot pucks at 90 mph:

a) What is the probability of a puck being shot at speeds in between 82 and 96 mph?
b) If Puck Chuck wanted to reset the machine so that only 17 in 1000 pucks would have velocities
of less than 80 mph, what would the mean output of the machine be set to?

Question 5

Determine the test statistic (Z* or t*) and the p-value for each of the each of the following situations and determine if they would cause the rejection of the null hypothesis if the confidence level was set at 95% in each case. (Hint: be wary of the sample size):

a) Ho: μ ≥ 33.8 g, Ha: μ < 33.8 g, sample mean = 30.9, sample standard deviation = 9, n = 36.
b) Ho: μ ≤ 145 km/h, Ha: μ > 145 km/h, sample mean = 159.5, s = 34, n = 42.
c) Ho: μ = 2.48 m, Ha: μ ≠ 2.48 m, sample mean = 2.99, s = 0.85, n = 16.
d) Ho: μ ≥ 68.0 minutes, Ha: μ < 68.0 minutes, sample mean = 67.3, s = 3.7, n = 26.
e) Ho: μ = 22.7 °C, Ha: μ ≠ 22.7 °C, sample mean = 23.1 °C, s = 1.24 °C, n = 32.

Question 6

Montreal Alouettes quarterback Anthony Calvillo passed for over 6000 yards in the 2004 season, becoming only the third player to do so during the regular season in the Canadian Football League. While discussing this exploit one day, a friend of yours claims that Calvillo's average passing yards per season exceeds 3000. Curious, you collect a random sample of passing yards from his career statistics and find the following:
Year Yards Year Yards
1996 2571 2004 6041
1997 2177 2006 4714
1999 2592 2007 3608
2001 3671 2009 4639
2002 5013 2010 4839

Please conduct a complete hypothesis test to determine if Anthony Calvillo's passing performance in a given season does indeed exceed 3000 yards. Conduct this test at the 97.5% confidence level and determine the p-value.

Question 7

Socially conscious investors screen out stocks from alcohol and tobacco companies in favour of more "socially conscious" enterprises. One popular measure of stock value is the P/E ratio (price to earnings ratio), with a high P/E indicating that the stock might be overpriced (pay more for less earnings). The stock index of all major stocks has a mean P/E ratio of 20.9. A random sample of 36 "socially conscious" stocks gave a P/E ratio of 18.1 with a standard deviation of 7.4. Determine, with 95% confidence, if the stocks from "socially conscious" companies are different from the mean P/E ratio of all other major stocks. Please conduct a complete hypothesis test and find the p-value.