**1. Comparing Z-scores**

Two students A and B did two different exams. A did Finance and his grade was 7.9. B did Microeconomics and his grade was 7.6 (both grades on a scale from 0-10). The mean grade of all students who did Finance was 7.4 with standard deviation 3; the mean grade of all students who did Microeconomics was 7.5 with standard deviation 0.5.

Who performed better, A or B? Explain your answer.

**Solution.**

The probability that a student gets a grade higher than 7.9 for Finance is:

The probability that a student gets a grade higher than 7.6 for Microeconomics is:

Since it appears more difficult for B to get a 7.6 than it was for A to get a 7.9. So, although B has a lower grade than A, he performed better.

**2. Tulips**

A certain type of Amsterdam tulip bulb germinates 90% of the time. A backyard farmer planted 25 tulip bulbs. Answer the following questions.**a.** What is the probability that exactly 20 germinate?**b.** What is the probability that 20 or more germinate?**c.** What is the expected number of bulbs that germinate?

**Solution.****a.** Here we have a binomial distribution problem with . Use the binomial distribution formulas in Excel to calculate the probability that exactly 20 bulbs germinate (20 successes). The answer is 0.0646 or 6.5%.**b.** or 96.7% by using the cumulative binomial distribution.**c.** It is obvious that the expected (average) number of bulbs that germinate equals .

**3. Incoming faxes**

After an analysis of incoming faxes the manager of an accounting firm determined the probability distribution of the number of pages per fax as follows:

x | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

P(x) | 0.05 | 0.12 | 0.20 | ? | 0.15 | 0.10 | 0.08 |

**a.** Compute the probability that the number of pages equals 4?**b.** How many percent of the faxes has more than 5 pages?**c.** Compute the mean and the variance of the number of pages per fax.

**Solution.****a.** , so .**b **. So the percentage is 18%.

c.

**4. National railroad**

The Dutch National Railroad *NS* found that 85% of all trains arrive on time.**a.** If you randomly select 100 train arrivals, compute the probability that exactly 85 of these trains arrive on time.**b.** Compute the probability that 85 trains or more arrive on time.**c.** Under certain conditions a binomial distribution may be approximated by a normal distribution. What are the mean and standard deviation of the normal distribution that approximates the binomial distribution described above.**d.** Repeat the question under b., now using the normal distribution. Explain the difference.

**Solution.****a.** This is a binomial experiment with and (probability of being on time). If denotes the number of trains on time, Excel computes .**b.** We compute ..**c.** The mean of this binomial experiment is and which can be used as the parameters of the approximating normal distribution .**d.** Taking the correction factor for continuity into account we get as approximation .

**5. Roulette**

In a game of roulette, a steel ball is rolled onto a wheel that contains 18 red, 18 black, and 2 green slots. If the ball is rolled 25 times, find the probability of the following events.**a.** The ball falls into the green slots two or more times.**b.** The ball does not fall into any green slots.**c.** The ball falls into black slots more than 5 and less than 15 times.

**Solution.****a.** There are 38 slots in total . The probability that the ball falls in a green slot (success) is . So, we have a binomial experiment with the total number of experiments and . Excel only computes the probability . So . (Naturally, this relatively simple calculation can also be done manually).**b.** In this case we have , (ball not in green slots) and we compute the probability that 25 times out of 25 experiments the ball does not fall in the green slots:

.**c.** In this case we have , P(ball in black slot) . We have to compute:

, where we used Excel.

**6. Height Dutch women**

The heights of Dutch women are normally distributed with a mean of 170.7 cm and a standard deviation of 6.3 cm.**a. **What is the probability that a randomly selected woman is taller than 178 cm?**b. **A random sample of four women is selected. What is the probability that the sample mean height is greater than 178 cm.**c.** A random sample of 100 women is selected. What is the probability that the sample mean height is greater than 178 cm.

**Solution**.

a. Use Excel (Cumulative Distribution Function) to compute .

b. If is the mean height of 4 women, then , which can be computed either manually (table) or by Excel.

c. Now the probability is almost.