Normal distribution

A group of homeowners are investigating the level of lead in their tap water.

Five homeowners each collected a 100 ml sample of their water at the same time each morning for 10 consecutive days.

All samples were sent to the same laboratory where the level of lead in each sample was measured.

Given these control measures, comment on the reliability of their results.

In previous tests in this area, the level of lead in the water was found to be normally distributed.

The mean level of lead was 7 µg/l and the upper action limit was 9.4 µg/l

Find the standard deviation used to set the upper action limit.

“ There are strict regulations concerning the amount of lead allowed in drinking water. The current drinking water standard is no more than 10 µg/ l ” (Source: www.stwater.co.uk)

One homeowner claims that the lead content in their tap water often exceeds 10 µg/l

Explain why this homeowner is likely to be wrong.

Control charts for the sample means and the sample ranges for the five homeowners over the ten days are shown below.

Decide what actions, if any, should be taken. Justify your answer by referring to both the sample means and the sample ranges.

The wingspans of a species of butterfly are normally distributed. The mean length of the wingspans for this species of butterfly is 8.2 cm. Only 2.5% of this species of butterfly have a wingspan longer than 8.6 cm.

Show that the standard deviation of this species of butterfly is 0.2 cm.

On the axis below, sketch a diagram for this normal distribution.

Sarah claims to have caught a butterfly of this species with a wingspan of 7.5 cm.

Comment on the likelihood of her claim being true. Give a reason for your answer.

The lifespan of this butterfly species is known to be 15 days or less.

Liam wants to know if the lifespan of this butterfly species can be modelled by a normal distribution.

He has collected data for 34 random butterflies of this species. First, Liam tabulates this data. He produces this table to show the length of time each butterfly lived.

Length of time lived (d days)	Frequency
0 d 3	4
3 d 6	9
6 d 9	12
9 d 12	7

Liam then produces this plan to check if the lifespan of this butterfly species can be modelled by a normal distribution.

Comment on whether or not Liam’s tabulation and plan are appropriate.

Roxann wants to investigate the resting heart rates for members of her running club.

She believes that the resting heart rates will be normally distributed.

Here is the plan for her investigation.

Discuss whether Roxann’s plans for collecting and presenting data are appropriate.

The mean of the daily winter temperatures and the mean of the daily summer temperatures in the UK have been recorded for each year from 1910 to 2018

This information has been used to model the two distributions of the mean temperatures.

The graphs below give information about these models.

Write down the name of the distribution that is suggested by each of these graphs.

Comment on the difference between the means of these two distributions.

The standard deviation for the distribution of the summer temperatures is 0.8 °C, correct to one decimal place.

Using the graph for the winter temperatures, calculate an estimate for the standard deviation of the distribution of the winter temperatures, correct to one decimal place.

The maximum height of the graph above the temperature axis for the winter temperatures is less than the maximum height of the graph above the temperature axis for the summer temperatures.

Explain why.

Carol says that the standard deviation of daily temperatures in summer for the years from 1910 to 2018 will also be 0.8 °C.

Explain whether or not Carol is correct.

Greta assumes that the means of the daily temperatures each year are independent.

She works out the probability that the means of the daily summer temperatures are greater than 15.6 °C for two consecutive years.

She concludes that this probability is less than 0.001

Using the model for the daily summer temperatures, assess Greta’s conclusion.

Kate is investigating the lengths of earthworms. She wants to find out if the lengths of common earthworms can be modelled by a normal distribution. She has found out that the greatest length of the common earthworm is 36 cm. (Source: www.nationalgeographic.com)

Kate plans to group the data that she will collect in the table shown below.

Comment on whether Kate’s choice of class intervals in the grouped frequency table is appropriate.

Bien is researching the lengths of time, in days, lived by flies. He wants to know if the lengths of time lived by flies can be modelled by a normal distribution.

He has collected data for 80 flies.

The table gives information about the length of time lived by each fly.

Here is Bien’s plan to test whether the lengths of time lived by flies can be modelled by a normal distribution.

Comment on whether Bien’s plan is appropriate.

Some students at a school walk home and some students go home by bus.

The times taken by the students at the school to walk home have a mean of 25 minutes and a standard deviation of 6 minutes.

The times taken to walk home can be modelled by a normal distribution.

Shanaya says,

“More than 80% of the students who walk home take between 19 and 37 minutes”

Use statistical calculations to assess Shanaya’s conclusion.

The diagram below shows a sketch of the distribution of the times taken by students to go home by bus.

Explain why it is not appropriate to use a normal distribution to model the times taken by students to go home by bus.

On the same grid, sketch a diagram showing the distribution of the times taken by students to walk home.

A is the event that a student walks to school, where P(A) = 0.6

B is the event that a student is driven to school, where P(B) = 0.3

Some students are driven some of the way to school and then they walk the rest of the way to school, where P(A and B) = 0.15

A student is picked at random from the students at the school.

Work out P(A or B).