when is known and n (large)
Foundation Year Program
2016-17
Rustem Iskakov
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Confidence interval and Hypothesis testing for population mean (µ) when is known and n (large) презентация
Содержание
- 1. Confidence interval and Hypothesis testing for population mean (µ) when is known and n (large)
- 2. Lecture overview: Learning outcomes At the end
- 3. Lecture overview: Learning outcomes At the end
- 4. Textbook Reference The content of this lecture
- 5. Terminology A range of values constructed
- 6. Terminology Probability of including the true value
- 7. Terminology Two extreme measurements within which an
- 8. Estimation of population parameters Point estimate
- 9. Point estimate VS Interval estimate Point estimate
- 10. Point estimate VS Interval estimate Point estimate
- 11. Point estimate VS Interval estimate Foundation Year
- 12. Point estimate VS Interval estimate Point estimate
- 13. Point estimate VS Interval estimate Point estimate
- 14. 7.5.1 Calculate and interpret confidence intervals for
- 15. The general formula for all confidence intervals
- 16. 7.5.1 Calculate and interpret confidence intervals for
- 17. Empirical rule Foundation Year Program 2016-17
- 18. Empirical rule Foundation Year Program 2016-17
- 19. 95% Confidence Interval of the Mean Bluman, Chapter 7 Foundation Year Program 2016-17 ? ?
- 20. Common Levels of Confidence Bluman, Chapter 7 Foundation Year Program 2016-17
- 21. Formula for the Confidence Interval of the
- 22. 7.5.1 Calculate and interpret confidence intervals for
- 23. 7.5.1 Calculate and interpret confidence intervals for
- 24. 7.5.1 Calculate and interpret confidence intervals for
- 25. 7.5.1 Calculate and interpret confidence intervals for
- 26. Hypothesis testing as well as estimation is
- 27. In Hypothesis testing beside sample statistics level
- 28. The level of significance, α, is a
- 29. In Hypothesis testing we compare a sample
- 30. 1. Hypothesis testing can be used to
- 31. 3. The alternative hypothesis, denoted
- 32. Types of Hypothesis testing Null Hypothesis (H0)
- 33. Step 1. Develop the null and alternative
- 34. Foundation Year Program 2016-17 Steps of Hypothesis
- 35. p-Value Approach to One-Tailed Hypothesis Testing
- 36. Critical Value Approach to One-Tailed Hypothesis
- 37. One-tailed test (left-tailed) Foundation Year Program 2016-17
- 38. One-tailed test (right-tailed) Foundation Year Program 2016-17
- 39. Two-tailed test Foundation Year Program 2016-17
- 40. 7.5.2 Test the hypothesis for a mean
- 41. Foundation Year Program 2016-17 Example 3
- 42. Foundation Year Program 2016-17 Example 3
- 43. Foundation Year Program 2016-17 Example 4
- 44. Foundation Year Program 2016-17 Example 4
- 45. Foundation Year Program 2016-17 Example 4
- 46. Foundation Year Program 2016-17 Example 4
- 47. Foundation Year Program 2016-17 Example 4
- 48. 7.5.2 Test the hypothesis for a mean
- 49. Foundation Year Program 2016-17 Example 5
- 50. Foundation Year Program 2016-17 Example 5
- 51. Foundation Year Program 2016-17 Example 5
- 52. 7.5.3 Test the hypothesis for the difference
- 53. 7.5.3 Test the hypothesis for the difference
- 54. 7.5.3 Test the hypothesis for the difference
- 55. Foundation Year Program 2016-17 Example 6
- 56. Foundation Year Program 2016-17 Example 6 1.6449
Lecture overview: Learning outcomes At the end of this lecture you should be able to: 7.6.1 Calculate and interpret confidence intervals for a population parameter 7.6.2 Test the hypothesis
Слайд 1NUFYP Mathematics & Computing Science
Pre-Calculus Course
L21 Confidence interval and
Hypothesis
testing for population mean (µ)
when is known and n (large)
when is known and n (large)
Слайд 2Lecture overview: Learning outcomes
At the end of this lecture you should
be able to:
7.6.1 Calculate and interpret confidence intervals for a population parameter
7.6.2 Test the hypothesis for a mean of a normal distribution,
Ho: µ=k,
H1: µ≠k or µ>k or µ
7.6.1 Calculate and interpret confidence intervals for a population parameter
7.6.2 Test the hypothesis for a mean of a normal distribution,
Ho: µ=k,
H1: µ≠k or µ>k or µ
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Слайд 3Lecture overview: Learning outcomes
At the end of this lecture you should
be able to:
7.6.3 Test the hypothesis for the difference between means of two independent normal distributions
Ho: µx - µy=0,
H1: µx - µy≠0 or µx - µy<0 or µx - µy>0
7.6.3 Test the hypothesis for the difference between means of two independent normal distributions
Ho: µx - µy=0,
H1: µx - µy≠0 or µx - µy<0 or µx - µy>0
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Слайд 4Textbook Reference
The content of this lecture is from the following textbook:
Chapter
3
Statistics 3 Edexcel AS and A Level Modular Mathematics S3 published by Pearson Education Limited
ISBN 978 0 435519 14 8
Further examples can be found in the textbook.
Statistics 3 Edexcel AS and A Level Modular Mathematics S3 published by Pearson Education Limited
ISBN 978 0 435519 14 8
Further examples can be found in the textbook.
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Слайд 5Terminology
A range of values constructed so that there is a
specified probability of including the true value of a parameter within it
CONFIDENCE INTERVAL
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Слайд 6Terminology
Probability of including the true value of a parameter within a
confidence interval
Percentage
Percentage
CONFIDENCE LEVEL
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Слайд 7Terminology
Two extreme measurements within which an observation lies
End points of the
confidence interval
Larger confidence – Wider interval
Larger confidence – Wider interval
CONFIDENCE LIMITS – CRITICAL VALUES
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Слайд 8Estimation of population parameters
Point estimate Interval estimate
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We have covered this
in previous lecture
Слайд 9Point estimate VS Interval estimate
Point estimate
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But, sample mean is still an approximation, and how close (ERROR) it is to true population mean value we do not consider in the Point estimate.
Слайд 10Point estimate VS Interval estimate
Point estimate
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Interval estimate
Слайд 11Point estimate VS Interval estimate
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Interval estimate
In interval estimate we
do consider ERROR
Interval estimate is a range of numbers around the point estimate within which the parameter is believed to fall
Слайд 12Point estimate VS Interval estimate
Point estimate
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Interval estimate
Until now we didn’t specify what is meant by error
Слайд 13Point estimate VS Interval estimate
Point estimate
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Interval estimate
Standard error
Слайд 147.5.1 Calculate and interpret confidence intervals for a population parameter
Interval estimate
provides us interval within which we believe value of true population mean falls
Then by using Standard Normal Distribution we can consider specific level of confidence that µ is really there by adjusting critical coefficient
Then by using Standard Normal Distribution we can consider specific level of confidence that µ is really there by adjusting critical coefficient
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Слайд 15The general formula for all confidence intervals are:
Point Estimate ± (Critical
Value) (Standard Error)
7.5.1 Calculate and interpret confidence intervals for a population parameter
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Слайд 167.5.1 Calculate and interpret confidence intervals for a population parameter
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Critical values
Слайд 21Formula for the Confidence Interval of the Mean for a Specific
α
Bluman, Chapter 7
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For a 90% confidence interval:
For a 99% confidence interval:
For a 95% confidence interval:
Слайд 227.5.1 Calculate and interpret confidence intervals for a population parameter
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Example 1
Слайд 237.5.1 Calculate and interpret confidence intervals for a population parameter
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Example 1
Слайд 247.5.1 Calculate and interpret confidence intervals for a population parameter
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Example 2
Слайд 257.5.1 Calculate and interpret confidence intervals for a population parameter
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Example 2
Слайд 26Hypothesis testing as well as estimation is a method used to
reach a conclusion on population parameter by using sample statistics.
7.5.2 Test the hypothesis for a mean of a normal distribution
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Hypothesis testing
Слайд 27In Hypothesis testing beside sample statistics level of significance (α) is
used to make a meaningful conclusion.
7.5.2 Test the hypothesis for a mean of a normal distribution
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Hypothesis testing
Слайд 28The level of significance, α, is a probability and is, in
reality, the probability of rejecting a true null hypothesis.
Confidence level
C = (1- α)
Level of Significance
α = 1 - C
Confidence level
C = (1- α)
Level of Significance
α = 1 - C
Level of Significance (α)
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Слайд 29In Hypothesis testing we compare a sample statistic to a population
parameter to see if there is a significant difference.
7.5.2 Test the hypothesis for a mean of a normal distribution
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Hypothesis testing
Слайд 301. Hypothesis testing can be used to determine
whether a statement
about the value of a
population parameter should or should
not be rejected.
population parameter should or should
not be rejected.
2. The null hypothesis, denoted by H0 , is a tentative
assumption about a population parameter.
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7.5.1 Calculate and interpret confidence intervals for a population parameter
Hypothesis testing
Слайд 31 3. The alternative hypothesis, denoted by Ha, is the
opposite of what is stated in the null hypothesis.
4. The hypothesis testing procedure uses data
from a sample to test the two competing
statements indicated by H0 and Ha.
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Hypothesis testing
7.5.1 Calculate and interpret confidence intervals for a population parameter
Слайд 32Types of Hypothesis testing
Null Hypothesis (H0)
Alternative Hypothesis (Ha or
H1)
Each of the following statements is an example of a null hypothesis and corresponding alternative hypothesis.
Each of the following statements is an example of a null hypothesis and corresponding alternative hypothesis.
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Слайд 33Step 1. Develop the null and alternative hypotheses.
Step 2. Specify the
level of significance α.
Step 3. Collect the sample data and compute the value of the test statistic.
p-Value Approach
Step 4. Use the value of the test statistic to compute the
p-value.
Step 5. Reject H0 if p-value < a.
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Steps of Hypothesis Testing
Слайд 34Foundation Year Program
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Steps of Hypothesis Testing
Critical Value Approach
Step 4. Use the
level of significance to determine the critical value and the rejection rule.
Step 5. Use the value of the test statistic and the rejection
rule to determine whether to reject H0.
Слайд 35p-Value Approach to
One-Tailed Hypothesis Testing
Reject H0 if the p-value
< α .
The p-value is the probability, computed using the
test statistic, that measures the support (or lack of
support) provided by the sample for the null
hypothesis.
If the p-value is less than or equal to the level of
significance α, the value of the test statistic is in the
rejection region.
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Слайд 36Critical Value Approach to
One-Tailed Hypothesis Testing
The test statistic
z has a standard normal probability
distribution.
distribution.
We can use the standard normal probability
distribution table to find the z-value with an area
of α in the lower (or upper) tail of the distribution.
The value of the test statistic that established the
boundary of the rejection region is called the
critical value for the test.
The rejection rule is:
Lower tail: Reject H0 if z < -zα
Upper tail: Reject H0 if z > zα
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Слайд 407.5.2 Test the hypothesis for a mean of a normal distribution,
Ho: µ=k, H1: µ≠k or µ>k or µ
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Example 3
Слайд 487.5.2 Test the hypothesis for a mean of a normal distribution,
Ho: µ=k, H1: µ≠k or µ>k or µ
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Example 5
Слайд 527.5.3 Test the hypothesis for the difference between means of two
independent normal distributions
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Слайд 537.5.3 Test the hypothesis for the difference between means of two
independent normal distributions
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Слайд 547.5.3 Test the hypothesis for the difference between means of two
independent normal distributions
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Example 6
