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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.
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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.
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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
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Commonly used z – values for Confidence Intervals
Confidence Interval z
90% 1.645
95% 1.96
98% 2.326
99% 2.576
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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
,
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Practice Questions
Statistics 3 and 4 by Jane Miller
Page 50, Exercise 3A
Question 1 onwards
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Statistics 3
Confidence Intervals and the t - Distribution
- Lesson 2 -
Key Learning Points/Vocabulary:
• The central limit theorem (S2)
• Unbiased estimate of the population variance (S2).
• Calculating the confidence interval for a large sample.
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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
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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 ,
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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.
Source: Page 48 of Statistics 3 by Jane Miller
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Practice Questions
Statistics 3 and 4 by Jane Miller
Page 50, Exercise 3B
Question 1 onwards
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Statistics 3
Confidence Intervals and the t - Distribution
- Lesson 3 -
Key Learning Points/Vocabulary:
• 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.
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Lesson 3 - Example Question
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?
Source: Page 55 of Statistics 3 by Jane Miller
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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 ,
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Practice Questions
Statistics 3 and 4 by Jane Miller
Page 50, Exercise 3C
Question 3 onwards
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Statistics 3
Confidence Intervals and the t - Distribution
- Lesson 4 -
Key Learning Points/Vocabulary:
• 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.
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Source: http://en.wikipedia.org/wiki/File:Student_densite_best.JPG
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Lesson 4 - Example Question
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
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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
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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 ,
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Practice Questions
Statistics 3 and 4 by Jane Miller
Page 62, Exercise 3D
Question 2 onwards
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Statistics 3
Confidence Intervals and the t - Distribution
- Lesson 5 -
Key Learning Points/Vocabulary:
• Hypothesis test on the population mean for a small sample from a normal population.
• Shortened name: t-Test.
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Lesson 5 - Example Question
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
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Practice Questions
Statistics 3 and 4 by Jane Miller
Page 67, Exercise 3E
Question 1 onwards
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Statistics 3
Confidence Intervals and the t - Distribution
- Lesson 6 -
Key Learning Points/Vocabulary:
• Mixed questions on Confidence Intervals and the t-Distribution.
• Mind map to summarise key learning points.
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Practice Questions
Statistics 3 and 4 by Jane Miller
Page 687, Miscellaneous Exercise 3
Questions 1, 3, 6 and 9 (first part only)