1. Estimations from Sample Data
Up to know, we’ve been working with a known population mean or proportion and then looked at sampling data from that population. Now we’re going to go the opposite way. We’re going to use sampling data to make decisions about the unknown value of the population mean or proportion.
In chapter 7, we had access to the population means and used that information to make probability statements about the individual x values taken from the population.
Then we had access to the population mean and made probability statements about the means of samples taken from the population (ch. 8).
Now we want to make probability statements about the unknown value of the population mean. Once again, we’ll rely on the CLT.

2. Point Estimates Population Parameter Mean, µ 𝑥 Variance, σ 2
Unbiased estimator
Formula
Mean, µ 𝑥
Variance, σ 2
𝑠 2
Proportion, π p
Remember, the expected value of the sample mean is the population mean and the expected value of the sample proportion is the population proportion. X bar and p are called unbiased estimators of the population mean and proportion. They’re called unbiased because they’re the same as the value they are used to estimate; that is, they don’t consistently induce a biased estimate. Because we use n-1 in the divisor, the variance is not a true unbiased estimator, but it will be close enough for us and we will use it as such.

3. Interval Estimates
Interval estimate for the mean describes a range of values that is likely to include the population mean.
Interval limits – the lower and upper values of the interval estimate
Confidence interval – An interval for which there is a specified degree of certainty that the actual population parameter will fall with the interval
Margin of error
When we know the population mean and s.d., we used standard normal to determine the proportion of sample means that will fall within a given number of standard error units form the known population mean. It is typical of inferential statistics for us to use the mean and standard deviation of a single sample as out best estimate of the unknown population mean and s.d.
If we establish the sample mean as the midpoint of an interval estimate for the population mean, the resulting interval may or may not include the actual value of the population mean.

4. Interval Estimates
Confidence coefficient/Confidence level – express the degree of certainty that an interval will include the actual value of the population parameter.
Coefficients are expressed as a number between 0 and 1 (.95)
Levels are expressed as percentages (95%)
Accuracy – the difference between the sample statistic and the actual parameter
Sometimes called sampling error.
Consider the following confidence interval: We are 90% confident that the population mean is greater than 100 and less than 200.
Some people think this means there is a 90% chance that the population mean falls between 100 and 200. This is incorrect. Like any population parameter, the population mean is a constant, not a random variable. It does not change. The probability that a constant falls within any given range is always 0.00 or 1.00.
The confidence level describes the uncertainty associated with a sampling method. Suppose we used the same sampling method to select different samples and to compute a different interval estimate for each sample. Some interval estimates would include the true population parameter and some would not. A 90% confidence level means that we would expect 90% of the interval estimates to include the population parameter; A 95% confidence level means that 95% of the intervals would include the parameter; and so on.

5. Interval Estimates - Example
In a random sample of 2000 households, the average income is $65,000 with a standard deviation of $12,000. Based on these data, we are 95% confident the population mean is between $64,474 and $65,526.
Point estimate of µ = 65,000
Point estimate of σ = 12,000
Interval estimate of µ =
64,474 to 65,526
Lower and upper interval limits for µ = 64,474 and 65,526
Confidence coefficient = 0.95
Confidence level = 95%
Accuracy: For 95% of such intervals, the sample mean would not differ from the actual population mean by more than $526
Confidence coefficient and confidence interval are, for all practical purposes, the same thing.

6. Special election 2010 Poll n Error +/- Coakley Brown 39.5 46.5 A 804
3.5%
43 %
52% B
574
4.2%
42%
52% C
1231
2.8%
46%
51% D
600
4.1%
45%
52% E
500
4.5%
48%
48%
Result
Coakley 47.1%
Brown 51.9%
Source: Real Clear Politics, 19 Jan 2010:

7. Example
Problem 9.11: In surveying a simple random sample of employed adults, we found that 450 individuals felt they were underpaid by at least $3000. Based on these results, we have 95% confidence that the proportion of employed adults who share this sentiment is between and
What is the point estimate for the population proportion?
What is the confidence interval estimate for the population proportion?
What is the confidence level and the confidence coefficient?
What is the accuracy of the sample result?

8. Confidence Interval: s known
where x = sample mean ASSUMPTION:
s = population standard infinite population
deviation
n = sample size
z = standard normal score
for area in tail = a/2
This would be a most unusual situation: we know the population s.d., but we don’t know the population mean.
CLT helps us if: population is normally distributed OR sample size >=30
Introduce the alpha term.
In this situation, our confidence interval would be computed by x-bar +/- z times the standard error of the mean. n z x s × + – :
© 2008 Thomson South-Western

9. Confidence interval practice
A simple random sample of 25 is collected from a normally distributed population.
σ = 17.0
𝑥 = 342
Construct and interpret a 95% confidence interval for the population mean.
Construct and interpret a 99% confidence interval for the population mean.

10. W. S. Gossett worked for the Guiness brewery in the early 1900s
W. S. Gossett worked for the Guiness brewery in the early 1900s. Guiness would not allow Gossett to publish his research, so he published under the name “student.”

11. Confidence Interval: s unknown
t distribution is the probability distribution for the random variable, t:
The t distribution has a mean of 0, but its shape is determined by degrees of freedom.
Specifically, for this distribution, df = n-1
Confidence interval estimates for unknown mean or standard deviation.
T distribution is much like normal, but more spread out. As n becomes very large, t and z are interchangeable.
Show the t distribution is slightly shallower than the normal distribution.

12. Using the t Distribution Table
For a sample size of n = 15, what t values would correspond to an area centered at t = 0 and having an area beneath the curve of 95%?
For a sample size of n = 99, what t values would correspond to an area centered at t = 0 and having an area beneath the curve of 90%?
When you have s and not sigma use the t-table, unless n is very large (>100)

13. Confidence Intervals Using the t distribution
Like before, only using t instead of z
Where
Sample mean
Sample standard deviation
Sample size
t value corresponding to the desired level of confidence
Estimated standard error of the mean

14. Confidence Intervals - Example
A random sample of 90 employees has been selected from those working in a company. The average number of overtime hours last week was with a sample standard deviation of hours.
What is the 98% confidence interval for the population mean?
Determine the appropriate value of t: DF T
Then fill in the rest of the formula
If we only wanted 90% confidence, would our spread be greater or smaller.

15. Confidence Intervals - Practice
The service manager for Appliance Universe conducted a random sample of 50 service calls from last year’s records. The same mean is 25 minutes and the sample standard deviation is 10 minutes.
Construct a 95% confidence interval.

16. Confidence Interval - Proportion
Π unknown
σ 𝑝 = 𝑝(1−𝑝) 𝑛
Correction for finite population, when required
Then, back to business as normal

17. Confidence Interval - Proportion
A major metropolitan newspaper selected a simple random sample of 1,600 readers from their list of 100,000 subscribers. They asked whether the paper should increase its coverage of local news. Forty percent of the sample wanted more local news. What is the 99% confidence interval for the proportion of readers who would like more coverage of local news?
When the sample does not include at least 10 successes and 10 failures, the sample size will be too small to justify the estimation approach presented in the previous lesson. 

18. Sample Size - Mean
Let’s consider a case where the population standard deviation is known.
Where:
n = required sample size
z = z value for which ± z corresponds to the desired level of confidence
σ = known or estimated value of the standard deviation
e = maximum likely error that is acceptable
So far, we took the results from our random samples and constructed a confidence interval. Now let’s work the problem backwards; that is, let’s start with a confidence interval and figure out how large our sample size needs to be to reach that level of confidence.
Central to our discussion is the fact that the maximum likely sampling error is one half the width of the confidence interval.
For population s.d. You probably won’t know it. You can use estimates based on other studies, conduct a small-scale pilot test, or estimate based on 1/6th the range of the data values.
We can’t use t, because we don’t have our sample, yet, so we don’t have a value for s.

19. Sample Size - Example
The governor wants to know the average amount teenagers earn during their summer vacation. He wants to be 95% confident the sample mean is within $50 of the population mean. Standard deviation is estimated at $400.
What sample size is required to achieve the desired results?

20. Sample Size Mean - Practice
A package-filling machine has been found to have a standard deviation of 0.65 ounces. A random sample will be conducted to determine the average weight of product being packed by the machine.
To be 95% confident that the sample mean will not differ from the actual population mean by more than 0.1 ounces, what sample size is required?
Nine hundred (900) high school freshmen were randomly selected for a national survey. Among survey participants, the mean grade-point average (GPA) was 2.7, and the standard deviation was 0.4. What is the margin of error, assuming a 95% confidence level?
Suppose we want to estimate the average weight of an adult male in Houston County. We draw a random sample of 1,000 men from a population of 1,000,000 men and weigh them. We find that the average man in our sample weighs 180 pounds, and the standard deviation of the sample is 30 pounds. What is the 95% confidence interval.

21. Sample Size - Proportion
Required sample size for a population proportion.
𝑛= 𝑧 2 𝑝(1−𝑝) 𝑒 2
When we have no idea what the population proportion is, set p=0.5
When we think the proportion might fall within some range, chose the value closest to 0.5.

22. Sample Size Proportion - Practice
A tourist agency would like to determine the proportion of U.S. adults who have vacationed in Mexico and wants to be 95% confident the sampling error will be no greater than 3%.
What is the required sample size to satisfy these parameters?