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Statistics 201 – Lecture 23
Confidence Intervals
•
Re-cap
1. Estimate the population mean with sample mean
• Know sample mean is unbiased estimator for m
• Distribution of sample mean:
2. Can construct a C.I. For the population mean, m
Confidence Intervals
•
Re-cap
3. Confidence interval has two parts
4. Confidence level, C, gives the long term proportion of times
intervals contain the true parameter, based on repeated
samples…what does this mean???
Confidence Interval for m
(unknown standard deviation)
• Situation:
• Have a random sample of size n
• Data from a normal population
• Suppose value of the standard deviation is unknown
• Value of population mean is unknown
• Use 1-sample t-confidence interval
Example
• Mercury contamination of salmon poses a direct threat to our health
• A variety of salmon in BC rivers were studied to examine the level
of mercury contamination
• It is believed that mercury levels in excess of 1/2 part per million is
the unsafe level of mercury concentration in edible foods
• The mean concentration of 10 sampled salmon was .2627 ppm and
the sample standard deviation was 0.1279 ppm
• Find a 90% confidence interval for the mean
Example
Summary
• When to use normal-based confidence intervals
• When to use t-based confidence intervals
Significance Testing
• Significance (Hypothesis) testing is a statistical technique for testing
a conjecture about a population parameter
• Has 4 Main Steps:
•
•
•
•
Null and Alternate Hypotheses
Test Statistic
P-Value
Decision based on pre-specified error rate
Example
• Heights of one-year-old girls normally distributed with mean 30
inches and standard deviation of 1.2 inches
• Company claims taking 500 mg of Vitamin C makes the girls taller
1. Hypotheses
• Begin by making an assumption of no change or no difference
• This statement is called the null hypothesis (H0)
• Test will be designed to assess evidence against H0
• Hypothesis we suspect is true is called alternate hypothesis (H1)
• Assume H0 is true, collect data and see if there is evidence against
H0 and in favor of H1
Example
• Heights of one-year-old girls normally distributed with mean 30
inches and standard deviation of 1.2 inches
• Company claims taking 500 mg of Vitamin C makes the girls taller
• H0:
• H1:
2. Test Statistic
•
Significance test uses data in the form of a test statistic
•
Measures compatibility of the null hypothesis with the data
•
Base on 2 principles:
1. Estimate of the parameter that appears in the hypotheses
2. Measures distance of the estimate and the hypothesized value
•
When H0 is true, the estimate should be close to the parameter on
average
Example
• Suppose a random sample of 100 baby girls are given 500mg of
vitamin C daily for 1 year
• Mean height of the girls after 1 year is 32 inches
• What is distribution of sample mean if H0 true
• What is distribution of sample mean if H1 true
3. P-Value
•
Assume null hypothesis is true
•
The P-value is the probability of observing a test statistic as
extreme or more extreme than the value actually observed when
the null hypothesis is true
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What does a small p-value imply?
•
How small is small?
Example (continued)
• If the null hypothesis is true, the distribution of the sample mean is:
• What does extreme mean in this case?
• P-Value=
4. Conclusion
•
How small must the p-value be to reject the null hypothesis
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Must decide if value of test statistic gives evidence in favor of
alternate hypothesis
•
Would like the probability of observing such values to be small if H0
is true
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The significance level of a test is:
Example (continued)
• P-Value=
• Significance level:
• Decision:
• Conclusion:
Similar to a jury trial
•
•
H0: State of no change
H1: Condition believed to be true
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Not guilty
Guilty
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Collect data and compute test
statistic
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Collect evidence and present to
jury
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Compute p-value
•
Weigh evidence
•
Reject or do not reject null
hypothesis based on significance
level and p-value
•
Decide if guilty or not guilty
•
How do we interpret significance level
•
Common sig. Levels
•
Have we proven H0 is true or false?