Confidence Intervals And The T Distribution

Statistics 3

Confidence Intervals and the t - Distribution

- Lesson 1 -

Key Learning Points/Vocabulary:

• The concept of a confidence intervals (notes).

• Calculating a 95% confidence interval for a sample drawn from a normal population of known variance.

• Calculating any confidence interval.

Lesson 1 - Example Question I

The heights of the 1320 lower school students at Poole High School are normally distributed with mean μ and with a standard deviation of 10cm.

A sample of size 25 is taken and the mean height of the sample is found to be 161cm.

a.) Find the 95% confidence interval for the height of the students.

b.) If 200 samples of size 25 are taken with a 95% confidence interval being calculated for each sample, find the expected number of intervals that do not contain μ, the population mean.

Lesson 1 - Example Question II

The masses of sweets produced by a machine are normally distributed with a standard deviation of 0.5 grams. A sample of 50 sweets has a mean mass of 15.21 grams.

a.) Find a 99% confidence interval for μ, the mean mass of all sweets produced by the machine correct to 2dp.

b.) The manufacturer of the machine claims that is produces sweets with a mean mass of 15 grams, state whether the confidence interval supports this claim.

Source: Page 48 of Statistics 3 by Jane Miller

Commonly used z – values for Confidence Intervals

Confidence Interval z

90% 1.645

95% 1.96

98% 2.326

99% 2.576

Generalisation: Sample from a Normal Population

A 100(1 – α)% confidence interval of the population mean for a sample of size n taken from a normal population with variance σ2 is given by

where x is the sample mean and the value of z is such that Ф(z) = 1 – ½α.

n

zxn

zx

,

Practice Questions

Statistics 3 and 4 by Jane Miller

Page 50, Exercise 3A

Question 1 onwards

Statistics 3


- Lesson 2 -


• The central limit theorem (S2)

• Unbiased estimate of the population variance (S2).

• Calculating the confidence interval for a large sample.

Unbiased Estimate of the Population Variance

Given a sample of size n (n large) from a population of which the variance is unknown, we estimate the population variance s2 as detailed below:

2

22

1x

n

x

n

ns

Generalisation: Large Sample for any Population

Given a large sample (n>30) from any population, a 100(1 – α)% confidence interval of the population mean is given by

where x is the sample mean and the value of z is such that Ф(z) = 1 – ½α.

n

szx

n

szx ,

Lesson 2 - Example Question

On 1st September, 100 new light bulbs were installed in a building, together with a device that detailed for how long each light bulb was used. By 1st March, all 100 light bulbs had failed. The data for the recorded lifetimes, t (in hours of use), are summarised by Σt = 10500 and Σt2 = 1712500. Assuming that the bulbs constituted a random sample, obtain a symmetric 99% confidence interval for the mean lifetime of the light bulbs, giving your answer correct to the nearest hour.


Practice Questions


Page 50, Exercise 3B

Question 1 onwards

Statistics 3


- Lesson 3 -


• Expectation and variance of the binomial distribution.

• Conditions for normal approximation to the binomial.

• Calculating the approximate confidence interval for a population proportion from a large sample.


You are the manufacturer of tin openers to be used specifically by left handed people. A random sample of 500 people finds that 60 of them are left handed. What is the 95% confidence interval for this estimate of the proportion of people who are left handed?


Generalisation: Confidence Interval for a Proportion

Given a large random sample of size n from a population in which a proportion of members p has a particular attribute, the approximate confidence interval is given by:

n

qpzp

n

qpzp ss

sss

s ,

Practice Questions


Page 50, Exercise 3C

Question 3 onwards

Statistics 3


- Lesson 4 -


• Use flow chart to help decide when to use either the t or z distribution.

• The t – distribution.

• Calculating the confidence interval for a small sample drawn from a normal population of unknown variance.

Source: http://en.wikipedia.org/wiki/File:Student_densite_best.JPG

http://en.wikipedia.org/wiki/File:Student_densite_best.JPG

http://en.wikipedia.org/wiki/File:Student_densite_best.JPG


Ten university physics students independently conducted experiments to determine the value of g. They obtained the following results:

9.812 9.807 9.804 9.805 9.812

9.808 9.807 9.814 9.809 9.807

Calculate the 95% confidence limits for g, stating any assumptions made.

Source: Page 105 of Statistics 2 by M E M Jones

Theory

For a random sample from a normal population with

mean μ, the variable has a t distribution with

ν degrees of freedom, where ν = n – 1.

That is,

nS

X2

12~

nt

nS

X

Generalisation: t-distribution

Given a sample from a normal population of unknown variance, a 100(1 – α)% confidence interval for the population mean is given by

where x is the sample mean and the value of t is such that P(T ≤ t) = 1 – ½α for ν = n – 1 degrees of freedom.

n

stx

n

stx ,

Practice Questions


Page 62, Exercise 3D

Question 2 onwards

Statistics 3


- Lesson 5 -


• Hypothesis test on the population mean for a small sample from a normal population.

• Shortened name: t-Test.


The weights of eggs laid by a hen when fed on ordinary corn are known to be normally distributed with a mean of 32kg. When a hen was fed on a diet of vitamin enriched corn a random sample of 10 eggs was weighed and the following results (in grams) were recorded:

31, 33, 34, 35, 35, 36, 32, 31, 36, 37

Test, using a 5% significance level, the claim that the new diet has increased the mean weight of eggs laid by the hen by more than 1g.

Source: Page 152 or Statistics2 by MEM Junes

Practice Questions


Page 67, Exercise 3E

Question 1 onwards

Statistics 3


- Lesson 6 -


• Mixed questions on Confidence Intervals and the t-Distribution.

• Mind map to summarise key learning points.

Practice Questions


Page 687, Miscellaneous Exercise 3

Questions 1, 3, 6 and 9 (first part only)

Technology

Confidence Intervals And The T Distribution