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Chapter 6: Introduction to Inference. http://pballew.blogspot.com/2011/03/100-confidence-interval.html. Confidence Interval: Definition. Example: Confidence Interval 1.

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chapter 6 introduction to inference
Chapter 6: Introduction to Inference

http://pballew.blogspot.com/2011/03/100-confidence-interval.html

example confidence interval 1
Example: Confidence Interval 1

Suppose we obtain a SRS of 100 plots of corn which have a mean yield (in bushels) of x̅ = 123.8 and a standard deviation of σ = 12.3.

What are the plausible values for the (population) mean yield of this variety of corn with a 95% confidence level?

ci conclusion
CI conclusion

We are 95% (C%) confident that the population (true) mean of […] falls in the interval (a,b) [or is between a and b].

We are 95% confident that the population (true) mean yield of this type of corn falls in the interval (121.4, 126.2) [or is between 121.4 and 126.2 bushels].

example confidence interval 2
Example: Confidence Interval 2

An experimenter is measuring the lifetime of a battery. The distribution of the lifetimes is positively skewed similar to an exponential distribution. A sample of size 196 produces x̅ = 2.268. The standard deviation is known to be 1.935 for this distribution.

a) Find and interpret the 95% Confidence Interval.

b) Find and interpret the 90% Confidence Interval.

c) Find and interpret the 99% Confidence Interval.

example confidence interval 2 cont
Example: Confidence Interval 2 (cont)

We are xx% confident that the population mean lifetime of this batter falls in the interval (x,y).

example confidence level precision
Example: Confidence Level & Precision

The following are two CI’s having a confidence level of 90% and the other has a level of 95% level: (-0.30, 6.30) and

(-0.82,6.82).

Which one has a confidence level of 95%?

example confidence interval 2 cont1
Example: Confidence Interval 2 (cont)

An experimenter is measuring the lifetime of a battery. The distribution of the lifetimes is positively skewed similar to an exponential distribution. A sample of size 196 produces x̅ = 2.268 and s = 1.935.

a) Find the Confidence Interval for a 95% confidence level.

b) Find the Confidence Interval for the 90% confidence level.

c) Find the Confidence Interval for the 99% confidence level.

d) What sample size would be necessary to obtain a margin of error of 0.2 at a 99% confidence level?

example confidence bound
Example: Confidence Bound

The following is summary data on shear strength (kip) for a sample of 3/8-in. anchor bolds: n = 78, x̅ = 4.25, s = 1.30.

Calculate a lower confidence bound using a confidence level of 90% for the true average shear strength.

We are 90% confident that the true average shear strength is greater than ….

cautions
Cautions
  • The data must be an SRS from the population.
  • You need to know the sample size.
  • Be careful about outliers.
  • You are assuming that you know σ.
conceptual question
Conceptual Question

One month the actual unemployment rate in the US was 8.7%. If during that month you took an SRS of 250 people and constructed a CI to estimate the unemployment rate, which of the following would be true:

1) The center of the interval would be 0.087

2) The interval contains 0.087

3) A 95% confidence interval estimate contains 0.087.

4) If you took 100 SRS of 250 people each, 95% of the intervals would contain 0.087.

tests of significance
Tests of Significance

http://www.rmower.com/statistics/Stat_HW/0801HW_sol.htm

example significance test
Example: Significance Test

You are in charge of quality control in your food company. You sample randomly four packs of cherry tomatoes, each labeled 1/2 lb. (227 g). The average weight from your four boxes is 222 g.

  • Is the somewhat smaller weight simply due to chance variation?
  • Is it evidence that the calibrating machine that sorts cherry tomatoes into packs needs revision?
example hypothesis
Example: Hypothesis

Translate each of the following research questions into appropriate hypothesis.

1. The census bureau data show that the mean household income in the area served by a shopping mall is $62,500 per year. A market research firm questions shoppers at the mall to find out whether the mean household income of mall shoppers is higher than that of the general population.

2. Last year, your company’s service technicians took an average of 2.6 hours to respond to trouble calls from business customers who had purchased service contracts. Do this year’s data show a different average response time?

example hypothesis cont
Example: Hypothesis (cont)

Translate each of the following research questions into appropriate hypothesis.

3. The drying time of paint under a specified test conditions is known to be normally distributed with mean value 75 min and standard deviation 9 min. Chemists have proposed a new additive designed to decrease average drying time. It is believed that the new drying time will still be normally distributed with the same σ = 9 min. Should the company change to the new additive?

example significance test con
Example: Significance Test (con)

You are in charge of quality control in your food company. You sample randomly four packs of cherry tomatoes, each labeled 1/2 lb. (227 g). The average weight from your four boxes is 222 g. The packaging process has a known standard deviation of 5 g.

c) What is the test statistic?

procedure for hypothesis testing
Procedure for Hypothesis Testing

1. Identify the parameter(s) of interest and describe it (them) in the context of the problem.

2. State the Hypotheses.

3. Determine an appropriate α level.

4. Calculate the appropriate test statistic.

5. Find the P-value.

6. Reject H0 or fail to reject H0 and why.

7. State the conclusion in the problem context.

The data does [not] give strong support (P-value = [x]) to the claim that the [statement of Ha in words].

example significance test cont
Example: Significance Test (cont)

You are in charge of quality control in your food company. You sample randomly four packs of cherry tomatoes, each labeled 1/2 lb. (227 g). The average weight from your four boxes is 222 g. The packaging process has a known standard deviation of 5 g.

d) Perform the appropriate significance test at a 0.05 significance level to determine if the calibrating machine that sorts cherry tomatoes needs to be recalibrated.

single mean test summary
Single mean test: Summary

Null hypothesis: H0: μ = μ0

Test statistic:

example ht vs ci
Example: HT vs. CI

You are in charge of quality control in your food company. You sample randomly four packs of cherry tomatoes, each labeled 1/2 lb. (227 g). The average weight from your four boxes is 222 g. The packaging process has a known standard deviation of 5 g.

e) Determine the 95% CI.

f) How do the results of part d) and e) compare?

example ht vs ci 2
Example: HT vs. CI (2)

Suppose we are interested in how many credit cards that people own. Let’s obtain a SRS of 100 people who own credit cards. In this sample, the sample mean is 4 and the sample standard deviation is 2. If someone claims that he thinks that μ > 2, is that person correct?

a) Construct a 99% lower bound for μ.

b) Perform an appropriate hypothesis test with significance level of 0.01.

c) How would the conclusion have changed if Ha: µ < 2?

example ht vs ci 21
Example: HT vs. CI (2)

b) The data does give strong support (P = 0) to the claim that the population average number of credit cards is greater than 2.

example ht vs ci 22
Example: HT vs. CI (2)

c) The data does not give strong support (P > 0.5) to the claim that the population average number of credit cards is less than 2.

example 1 homework
Example 1: Homework

A group of 15 male executives in the age group 35 – 44 have a mean systolic blood pressure of 126.07 and population standard deviation of 15.

  • Is this career group’s mean pressure different from that of the general population of males in this age group which have a mean systolic blood pressure of 128 at a significance level of 0.01?
  • Calculate and interpret the appropriate confidence interval.
  • Are the answers to part a) and b) the same or different? Explain your answer.
example 2 homework
Example 2: Homework

A new billing system will be cost effective only if the mean monthly account is more than $170. Accounts have a population standard deviation of $65. A survey of 41 monthly accounts gave a mean of $187.

  • Will the new system be cost effective at a significance of 0.05?
  • Calculate the appropriate confidence bound.
  • Are the answers to part a) and b) the same or different? Explain your answer.
  • What would the conclusion in part a) be if the monthly accounts gave a mean of $160? Please perform the hypothesis tests.
how small a p is convincing
How small a P is convincing?
  • What are the consequences of your conclusion?
  • Are you conducting a preliminary study?
notes for using p values
Notes for using P-values
  • There is no sharp border between “significant” and “not significant”
  • Do not use  = 0.05 as the default value!
statistical vs practical significance
Statistical vs. Practical Significance

An Illustration of the Effect of Sample Size on P-values

tests of significance1
Tests of Significance

http://www.rmower.com/statistics/Stat_HW/0801HW_sol.htm