Normal probabilities with the calculator Problem set B Worked solutions

The time a swimmer takes to swim 100 m is normally distributed with mean 64.0 s and standard deviation 2.8 s. Find the probability that she takes more than 68 s.

• \(X \sim N(64, 2.8^2)\)
• From the calculator: normal cdf with lower = 68, upper = 10^99, μ = 64, σ = 2.8
• P = 0.0766

The time to download a file is normally distributed with mean 7.5 seconds and standard deviation 2.1 seconds. Find the probability that a download takes less than 5 seconds.

• \(X \sim N(7.5, 2.1^2)\)
• From the calculator: normal cdf with lower = −10^99, upper = 5, μ = 7.5, σ = 2.1
• P = 0.117

The number of steps an office worker walks in a day is normally distributed with mean 8 200 and standard deviation 1 400. Find the probability that a worker walks between 7 000 and 10 000 steps in a day.

• \(X \sim N(8 200, 1 400^2)\)
• From the calculator: normal cdf with lower = 7 000, upper = 10 000, μ = 8 200, σ = 1 400
• P = 0.705

The mass, \(X\) grams, of a pebble from a beach is modelled by \(X \sim N(40, 25)\). Find the probability that a pebble has mass less than 33 g.

• \(\sigma = \sqrt{25} = 5\)
• From the calculator: normal cdf with lower = −10^99, upper = 33, μ = 40, σ = 5
• P = 0.0808

The length of a bolt is normally distributed with mean 18 mm and standard deviation 2.5 mm. Sketch the normal curve and shade the region for the probability that a bolt is shorter than 14 mm. Then find the probability.

• Sketch: bell curve centred at 18; mark 14 to the left of the centre and shade the area to its left
• From the calculator: normal cdf with lower = −10^99, upper = 14, μ = 18, σ = 2.5
• P = 0.0548

The box plot shows the number of goals scored in a season by 70 amateur players. A student says a normal distribution is a good model. Do you agree? Give a reason.

• Median − Q1 = 2 and Q3 − median = 4
• The upper whisker (14) is much longer than the lower (1)
• The data is skewed, but a normal distribution is symmetric, so it is not a good model.

The time that members of a cycling club take to ride 20 km is normally distributed with mean 41 minutes and standard deviation 5 minutes.

1. (a) Find the probability that a rider takes less than 35 minutes.

   • From the calculator: normal cdf with lower = −10^99, upper = 35, μ = 41, σ = 5
   • P = 0.115

2. (b) Find the probability that a rider takes between 38 and 50 minutes.

   • From the calculator: normal cdf with lower = 38, upper = 50, μ = 41, σ = 5
   • P = 0.690

The internal diameter of a pipe is modelled by \(X \sim N(20, 0.09)\) mm. A pipe is acceptable if its diameter is between 19.5 mm and 20.5 mm.

1. (a) Write down the standard deviation.

   • \(\sigma = \sqrt{0.09} = 0.3\)

2. (b) Find the probability that a pipe is acceptable.

   • From the calculator: normal cdf with lower = 19.5, upper = 20.5, μ = 20, σ = 0.3
   • P = 0.904

The diagram shows the normal distribution of the daily number of visitors to a museum, with mean 110 and standard deviation 12. The shaded region is under the curve.

1. (a) Write the probability that the shaded region represents.

   • The shaded region lies between 100 and 125
   • \(P(100 < X < 125)\)

2. (b) Find its value.

   • From the calculator: normal cdf with lower = 100, upper = 125, μ = 110, σ = 12
   • P = 0.692

A machine fills bottles of milk. The volume is normally distributed with mean 1 015 ml and standard deviation 9 ml. A bottle is accepted if its volume is between 1 000 ml and 1 030 ml.

1. (a) Find the probability that an item is accepted.

   • From the calculator: normal cdf with lower = 1 000, upper = 1 030, μ = 1 015, σ = 9
   • P(accepted) = 0.904

2. (b) Find the probability that an item is rejected.

   • Rejected is the complement: 1 − P(accepted), using the unrounded value
   • P(rejected) = 0.0956
   • Check: lower tail 0.0478 + upper tail 0.0478