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Statistics Assignment

?Business Analytics ?BUS1BAN – Topic 9 Estimation: Desc

•Business Analytics
•BUS1BAN – Topic 9
Estimation:
Describing a single population
Part 1
•Learning Objectives
LO1   Understand the fundamental concept of estimation
LO2   Understand the difference between a point estimator and an interval estimator
LO3   Understand the t distribution
LO4   Determine when to use the z-distribution and the t-distribution in estimation
LO5   Develop an interval estimate of a population mean when the population variance is known
LO6   Develop an interval estimate of a population mean when the population variance is unknown
LO7   Develop an interval estimate of a population proportion
LO8   Interpret interval estimates
LO9   Determine sample size for a given CI


•The story so far….
•The story so far….
We discussed data types and various methods of collecting data.
We learned about the graphical and numerical descriptive measures to summarize data in a meaningful way.
We learned about methods of assigning probabilities to events and the central and variability measures of discrete and continuous random variables.
We learned how, when and why statistical inference is used and considered the sampling distributions of the sample mean and sample proportion, and made probability statements about the sample statistics.
•Where to now?
When making such probability statements, we use some knowledge about the population parameter and the shape of the distribution of the population.
However, in almost all realistic situations population parameters are unknown.
Therefore, from now onwards, we will use the sampling distribution to draw inferences about the unknown population parameters.
•Statistical Inference: Introduction
•Statistical Inference: Introduction
Statistical inference is the process by which we acquire information about populations from samples.
There are two procedures for making inferences:
•Estimation (week 10)
•Hypothesis testing (week 11)
•Statistical Inference: Introduction
Estimation
Use estimation if we have no idea about the value of the population parameter being investigated
Hypothesis testing
Use hypothesis tests when we have some idea of the value of the population parameter being investigated, or if we have some hypothesised value against which we can compare our sample results.
•11.1 Estimation: Introduction
What do you think of when you hear estimation?
  Estimate the width of a property
  Estimate the height of a tree etc.

Now extend that to a sample of data and how we might estimate values of population parameters.
  Estimate the width of all properties in Bundoora
  Estimate the height of all trees on campus
  Anything missing?
•Estimation: More examples
•A bank conducts a survey to estimate the number of times customer will actually use ATM machines.
•A random sample of processing times used to estimate the mean production time and the variance of production time on a production line.
•A survey of eligible voters to gauge support for the federal government’s new carbon emissions reforms.
•Defining estimation..
• 
•Types of Estimation
There are two types of estimators:
•Point estimator
•Interval estimator
•Point Estimator
• 
•Interval Estimator
•Point vs Interval estimator
•Characteristics of an estimator
•Unbiased Estimators
An unbiased estimator of a population parameter is an estimator whose expected value is equal to that parameter.


•Unbiased Estimators…
• 
•Consistent estimators
An unbiased estimator is said to be consistent if the difference between the estimator and the parameter grows smaller as the sample size grows larger.
•Consistent estimators…
• 
•Efficient estimators
If there are two unbiased estimators of a (population) parameter, the one whose variance is smaller is said to be relatively efficient.




•Efficient estimators…
• 
•'Best' estimator
• 
•Estimation: Approach
We utilise the following approach for making statistical inferences about populations:
1.identify the parameter to be estimated
2.specify the parameter’s estimator and its sampling distribution
3.construct an interval estimator.
•11.2 Estimating the population mean m when the population variance s2 is known
• 
• 
•RECAP: Calculating Normal Probabilities
We can use the following function to convert any normal random variable to a standard normal random variable…
•RECAP: Central Limit Theorem…
• 
•RECAP: Sampling Distribution of the Sample Mean
•RECAP: Sampling Distribution of the Sample Mean
• 
•11.2 Estimating the population mean m when the population variance s2 is known
How is an interval estimator produced from a sampling distribution?
•And it follows a standard normal distribution.
•Estimating m when s2 is known
•Estimating m when s2 is known…
  For example, for a = 0.05 (a/2 = 0.025), the symmetry of the normal distribution with the sampling distribution of the sample mean leads to:
In general,
• 
The probability (1 − a) is called the level of confidence.

                       is called the lower confidence limit (LCL).

                       is called the upper confidence limit (UCL).




(1-a)100% confidence interval for m:
      or
This can also be written as
    or  ,  where  

B is half the width of the confidence interval. 
That is, 2B is the width of the confidence interval. 

Three commonly used confidence levels, (1 – a)
•Example 1
XM11-01 The sponsors of television shows targeted at children wanted to know the amount of time children spend watching television, since the types and number of programs and commercials presented are greatly influenced by this information. As a result, a survey was conducted to estimate the average number of hours Australian children spend watching television per week. From past experience, it is known that the population standard deviation σ is 8.0 hours.
The following are the data gathered from a sample of 100 children. Find the 95% confidence interval estimate of the average number of hours Australian children spend watching television.
•Example 1
•Example 1 - Solution
• 
•Example 1 - Solution
• 
•Example 1 - Solution
Calculating manually
Therefore, a 95% confidence interval estimator of μ is




We therefore estimate that the average number of hours children spend watching television each week lies somewhere between
  LCL = 25.62 hours and UCL = 28.76 hours
•Interpreting the results
That is, the average time spent watching TV by Australian children is between 25.6 hours and 28.8 hours. This type of estimate is correct 95% of the time.
As a consequence, a network executive may decide (for example) that, since the average child watches at least 25.6 hours of television per week, the number of commercials children see is sufficiently high to satisfy the programs’ sponsors.
Incidentally, the media often refer to the 95% confidence interval as ’19 out of 20 times’, which emphasises the long-run aspect of a confidence interval.
•Interpreting the interval estimator
Some people erroneously interpret the confidence interval estimate in this example (Example 1) to mean that there is a 95% probability that the population mean of the number of hours Australian children watch television lies between 25.623 and 28.759.
This interpretation is wrong because it implies that the population mean is a variable about which we can make probability statements.
In fact, the population mean is a fixed but unknown quantity. Consequently, we cannot interpret the confidence interval estimate of µ as a probability statement about µ.
•Interpreting the interval estimator
To translate the confidence interval estimate properly, we must remember that the confidence interval estimator was derived from the sampling distribution of the sample mean.
We used the sampling distribution to make probability statements about the sample mean.
Although the form has changed, the confidence interval estimator is also a probability statement about the sample mean.
•An illustrative application
• 
•An illustrative application
• 
•An illustrative application
• 
•An illustrative application
•An illustrative application
What went wrong with samples 5, 16, 22 and 34?
The answer is nothing. Statistics does not promise 100% certainty. In fact, in this illustration, we expected 90% of the 40 intervals to include μ and 10% to exclude μ.
Since we produced 40 confidence intervals, we expected that 4 (10% of 40) intervals would not contain μ = 3.5.
•Factors that determine the width of a confidence interval for m
• 
•Factors that determine the width of a confidence interval for m…
1.Level of confidence, 1-a. (B is directly proportional to a, via za/2)
 




As the level of confidence (1-a) increases from 0.90 to 0.99, za/2 also increases (from 1.645 to 2.575). Therefore, in general, when the level of confidence increases, the width also increases.

•Factors that determine the width of a confidence interval for m…
2.Population standard deviation, s.
  B is directly proportional to s and when s increases, the width (2B) also increases. For example, if s is doubled, then the width would also be doubled or if s is halved, then the width would be halved as well.
3.Sample size, n.
  B is inversely proportional to Ön. When n increases, the width (2B) decreases. For example, if n increases by 4 times, the width would be halved.

•The width of the confidence interval
A larger confidence level (1-a) produces a wider interval.
Larger values of standard deviation (s) produce wider confidence intervals.
Larger sample sizes (n) produces narrower confidence intervals.
•The width of the confidence interval
A wide interval provides little information.
For example, suppose we estimate with 95% confidence that an accountant’s average starting salary is between $15 000 and $100 000.
Contrast this with: a 95% confidence interval estimate of starting salaries between $42 000 and $45 000.
The second estimate is much narrower, providing accounting students more precise information about starting salaries.

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