Today: Statistics

A bit of revision of The Normal Distributionand starting a new chapter, Estimation

Question 1

What type of distribution is the normal distribution?

Question 2

What is special about the normal distribution curve?

Question 3

What does the area under the curve add up to?

Question 4

What does the z-value represent?

Question 5

What symbols are used for mean and standard deviation?

Question 6

Write down the standardising formula.

Question 7

We have a normal distribution X ~ N( 52, 42)Find:

a). P(X < 57)

b). P(X > 57)

c). P(X < 49)

d). P(49 < X < 57)

e)*. P(X = 57)

Question 8

The distance a football is kicked has a normal distributionX ~ N(70,82)How far would you need to kick to be:

a). in the top 10%?

b). In the top 5%?

c). in the bottom 5%

Question 9

We have the normal distribution X ~ N(35, σ2)24% are over 40.

Calculate the standard deviation, σ

Question 10*

In an examination, 10% of candidates scored more than 70 marksand 15% scored fewer than 35 marks. Find the mean and standarddeviation.

(hint: you will need to use the standardising formula twiceto get a pair of simultaneous equations)

Today – New Chapter Estimation for The Normal Distribution

Looking out how samples can give us estimates for the meanand standard deviation

If X ~ N(μ, ) then X ~ N(μ, )

However the standard deviation of the means is not the same as the standard deviation of the population!

eg. Using the sampling distribution and tables

The length of a particular species of worm is normally distributed with mean 5.6cm and standard deviation 0.4cm.

What is the probability that the mean length of a random sample of 12 worms is greater than 5.7cm?

Recap:

We saw last week that the mean of the samples can be used as an unbiased estimator for the population mean in the Normal Distribution.

X ~ N(μ, )

Remember from earlier,for the whole population it was X ~ N(μ, σ2)

σ2

n

Today – working out an unbiased estimate for the standard deviation/variance

In Chapter 1,

We had these formulae for calculating the standard deviation of a sample,

or

It will be useful to think about these today.

Up to now, we have been given the population standard deviation.

Just like the mean, we can estimate the standard deviation (√variance)from a sample too:

given in formula book

The speeds in km/hr of 10 cars entering a village were:

45 40 49 53 48 57 50 60 47 and 56

Calculate unbiased estimates for the mean and standard deviationof these speeds.

As with chapter 1, there is an alternative formula for calculating an estimate of the standard deviation:

or

A random sample of 30 bolts were taken from a production lineand the diameters were accurately measured.

It was found that Σd = 60.85 and Σd2 = 123.5

a). Calculate an unbiased estimate for the population mean

b). Which formula would you need to use for calculating an estimate for the standard deviation? Why?

c). Calculate an estimate for the standard deviation

d). Work out an estimate for the standard error

We are going to weigh 10 bags from our class:

a). Calculate an estimate for the population mean and standard deviation

b). Using the standardising formula and tables, work out P(X > )

c). What assumption have we made about the distribution of the school bags?

We are going to weigh 10 bags from our class:

a). Calculate an estimate for the population mean and standard deviation

b). Using the standardising formula and tables, work out P(X > )

c). What assumption have we made about the distribution of the school bags?

The Central Limit Theorem (CLT) says that if a random sample of size n is taken then

X ~ N(μ, )

even if the population isn’t normally distributed! Yay!

But the sample has to be bigger than around 30

If you’d like to know more about why this is, read P126-128 ofyour textbook.

σ2

n