The normal distribution and when to use it Problem set A One question at a time

A Galton board has 10 rows of pins. At each pin a ball goes left or right with probability 0.5, independently. Let B be the number of times a ball goes right, so \(B \sim B(10, 0.5)\).

1. (a) Find the mean and standard deviation of B.

   • \(np = 10 \times 0.5 = 5\)
   • \(\sigma = \sqrt{10 \times 0.5 \times 0.5} = 1.58\)

2. (b) Many balls are dropped. Describe the shape of the pile of balls and explain why it has that shape.

   • B is binomial with p = 0.5, so the pile is symmetric about the mean.
   • With many rows the columns follow a smooth bell shape: a normal curve.

The length of a salmon is normally distributed with mean 60 cm and standard deviation 4 cm. Use the 68–95–99.7 rule.

1. (a) Find the percentage between 52 and 68 cm.

   • \(52 = 60 - 2 \times 4\) and \(68 = 60 + 2 \times 4\)
   • These are 2 standard deviations either side of the mean: 95%

2. (b) Find the percentage below 56 cm.

   • \(56\) is 1 standard deviation below the mean
   • 68% is within 1 sd, so 32% is outside, 16% in each tail: 16%

3. (c) Find the percentage between 60 and 68 cm.

   • The mean to 2 sd above is half of the 95% within 2 sd
   • \(95 \div 2 = 47.5\%\)

The box plot shows the time in seconds that 80 web pages take to load. Is a normal distribution a suitable model? Give reasons.

• Median − Q1 = 0.6 but Q3 − median = 2
• The upper whisker (5.1) is much longer than the lower (0.6)
• A normal distribution is symmetric. This data is not, so a normal model is not suitable.

The box plot shows the masses in kg of 100 sheep. Is a normal distribution a suitable model? Give a reason.

• Median − Q1 = 5 and Q3 − median = 5
• The whiskers are 7 and 7
• The box plot is symmetric about the median with no outliers, so a normal model is reasonable.

The time, in minutes, that a customer waits in a queue is modelled by a normal distribution with mean 4.5 and standard deviation 2.8.

1. (a) How many standard deviations is the limit 0 below the mean?

   • \((4.5 - 0) \div 2.8 = 1.61\)

2. (b) Explain why a normal distribution is not a good model. Use your answer to part (a).

   • The limit is only 1.61 standard deviations below the mean.
   • From the calculator: normal cdf with lower = −10^99, upper = 0, μ = 4.5, σ = 2.8: 0.0540
   • That is a chance of impossible values, so the model is poor.

The mass of a jar of jam is normally distributed with mean 250 g and standard deviation 5 g.

1. (a) Use the 68–95–99.7 rule to give an interval that contains about 99.7% of the values.

   • \(250 \pm 3 \times 5\)
   • 235 to 265 g

2. (b) One jar measures 268 g. How many standard deviations from the mean is that? Is it unusual?

   • \((268 - 250) \div 5 = 3.60\)
   • More than 3 standard deviations from the mean: fewer than 0.3% of values are this far out. It is unusual.

A Galton board is tilted so that a ball goes right with probability 0.2 at each of its 10 rows of pins. Let B be the number of times a ball goes right, so \(B \sim B(10, 0.2)\).

1. (a) Find the mean and standard deviation of B.

   • \(np = 10 \times 0.2 = 2\)
   • \(\sigma = \sqrt{10 \times 0.2 \times 0.8} = 1.26\)

2. (b) How many standard deviations is 0 below the mean?

   • \(2 \div \sqrt{1.6} = 1.58\)

3. (c) Explain why the pile of balls is not a normal shape.

   • B cannot be below 0, and the mean is close to 0, so the left tail is cut off.
   • The right tail is longer: the pile is skewed, not a bell.