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

A pinball toy has 14 rows of pegs. At each peg a marble goes left or right with probability 0.5, independently. Let B be the number of times a ball goes right, so \(B \sim B(14, 0.5)\).

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

   • \(np = 14 \times 0.5 = 7\)
   • \(\sigma = \sqrt{14 \times 0.5 \times 0.5} = 1.87\)

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 volume of juice in a bottle is normally distributed with mean 250 ml and standard deviation 12 ml. Use the 68–95–99.7 rule.

1. (a) Find the percentage between 226 and 274 ml.

   • \(226 = 250 - 2 \times 12\) and \(274 = 250 + 2 \times 12\)
   • These are 2 standard deviations either side of the mean: 95%

2. (b) Find the percentage below 238 ml.

   • \(238\) 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 250 and 274 ml.

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

The box plot shows the number of messages that 90 students sent in one day. Is a normal distribution a suitable model? Give reasons.

• Median − Q1 = 3 but Q3 − median = 6
• The upper whisker (15) is much longer than the lower (2)
• A normal distribution is symmetric. This data is not, so a normal model is not suitable.

The box plot shows the times in seconds of 120 runners in a 100 m race. Is a normal distribution a suitable model? Give a reason.

• Median − Q1 = 3 and Q3 − median = 3
• The whiskers are 3 and 3
• The box plot is symmetric about the median with no outliers, so a normal model is reasonable.

The number of centimetres of rain that fall in a day in a desert town is modelled by a normal distribution with mean 1.6 and standard deviation 1.1.

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

   • \((1.6 - 0) \div 1.1 = 1.45\)

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

   • The limit is only 1.45 standard deviations below the mean.
   • From the calculator: normal cdf with lower = −10^99, upper = 0, μ = 1.6, σ = 1.1: 0.0729
   • That is a chance of impossible values, so the model is poor.

The diameter of a piston is normally distributed with mean 80 mm and standard deviation 0.4 mm.

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

   • \(80 \pm 3 \times 0.4\)
   • 78.8 to 81.2 mm

2. (b) One piston measures 81.5 mm. How many standard deviations from the mean is that? Is it unusual?

   • \((81.5 - 80) \div 0.4 = 3.75\)
   • More than 3 standard deviations from the mean: fewer than 0.3% of values are this far out. It is unusual.

A pinball toy is tilted so that a marble goes right with probability 0.15 at each of its 12 rows of pegs. Let B be the number of times a ball goes right, so \(B \sim B(12, 0.15)\).

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

   • \(np = 12 \times 0.15 = 1.8\)
   • \(\sigma = \sqrt{12 \times 0.15 \times 0.85} = 1.24\)

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

   • \(1.8 \div \sqrt{1.53} = 1.46\)

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.