Central Limit Theorem

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Transcript Central Limit Theorem

Central Limit Theorem
For a sample size n  30 taken from a nonnormal population with mean  and variance
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
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
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 5
A 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
Example 6
Lightning 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.