Central Limit Theorem

For a sample size n 30 taken from a non-normal population with mean and variance 2:

Significance testing

A pre-specified value to compare against a probability is called the significance level and is usually quoted as a percentage

e.g. we may say something is significant at the 5% level.

Example 1 Electric light bulbs have lifetimes that are

normally distributed with mean 1200 hours and standard deviation 150.

It is suspected that a batch is substandard. To test this a sample of 50 bulbs is taken.

a)Determine the significance level of a rule which would conclude that a batch is substandard if the sample mean lifetime is less than 1160hrs.

Example 1

(b) Determine the rule which would have a significance level of 1%.

(c) What conclusion should be drawn if the observed sample mean is 1150hrs and the chosen significance level is 1%?

Example 2

A machine filling milk cartons delivers mean amount of 500ml per carton, variance 70.3ml.

A sample of 50 cartons taken to check the machine has a mean amount of milk of 502.5ml.

Test, using a 5% significance level, whether the machine needs adjusting.

Example 3

A garden centre sells flower seeds that have a germination rate of 0.75. A packet of 20 seeds were sown. A new brand of seeds claims to have a higher germination rate.

a) Find the significance level of the rule that accepts the claim if X 18.

b) Find a rule whose significance level is 0.05.

c) What conclusion should be drawn if 19 seeds germinate?

Example 4

A coin is tossed 50 times to test if it's unbiased.

a) Find the significance level of the rule that the coin is biased if X 19 or X 31 where X is the number of heads.

b) Using a 5% significance level what conclusion should be drawn if 32 heads are obtained?

Example 5A traffic warden issues a mean number of 1.6

tickets per day. The Council decides to employ another to see if more tickets will be issued. They work 5 days.

a) The council decides that if the number of tickets was 12 or more they would conclude more tickets are issued. Find the significance level of this rule.

b) Given that the observed value was 13 find the p-value and deduce the minimum significance level for the conclusion to be that the mean number of tickets issued per weekday has increased.

Example 6Lightning strikes a church at a rate of 0.05 times

per week. In a ten year period the church was struck 35 times. The vicar claims there has been an increase in electrical storms. Test at a 5% significance level if he is correct.

Confidence Intervals

If I want the values (i.e. the interval) that, say, 95% of my data lies between I can use the Normal Distribution.

In general

A Confidence Interval (CI) is found by:

The critical value depends on the %:

Example 1

The pulse rate of 90 people was taken. The mean was 70 beats per minute and the standard deviation was 5 beats. Find

a) A 95% CI for the population mean

b) A 98% CI

c) 99% CI

d) 90% CI

e) 94% CI

Example 2

A plant produces steel sheets whose weights have a Normal Distribution with sd 2.1kg. A random sample of 49 sheets had mean weight 36.4kg. Find a 99% CI for the population mean.

Confidence Intervals for the difference between two means

In general,

The larger the sample the more accurate the CI. If they’re not Normal Variables but N is large we use the Central Limit Theorem.

Example