St at is t ic s 201 – Lec t ure 23

Conf idenc e In t erva ls

• Re-cap

1. Estimate the population mean with sample mean• Know sample mean is unbiased estimator for

• Distribution of sample mean:

2. Can construct a C.I. For the population mean,

Conf idenc e In t erva ls

• Re-cap3. 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???

Conf idenc e In t erva l for (unk now n st andard dev ia t ion)

• 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

Ex am ple

• 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

• I t 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

Ex am ple

Sum m ary

• When to use normal-based confidence intervals

• When to use t-based confidence intervals

Signi f ic anc e Test ing

• 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

Ex am ple

• 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. Hypot heses

• 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

Ex am ple

• 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 St at is t ic

• 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 hypotheses2. Measures distance of the estimate and the hypothesized value

• When H0 is true, the estimate should be close to the parameter on average

Ex am ple

• 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

• What does a small p-value imply?

• How small is small?

Ex am ple (c ont inued)

• If the null hypothesis is true, the distribution of the sample mean is:

• What does extreme mean in this case?

• P-Value=

4. Conc lus ion

• How small must the p-value be to reject the null hypothesis

• 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

• The significance level of a test is:

Ex am ple (c ont inued)

• P-Value=

• Significance level:

• Decision:

• Conclusion:

Sim i lar t o a jury t r ia l

• H0: State of no change• H1: Condition believed to be true

• Collect data and compute test statistic

• Compute p-value

• Reject or do not reject null hypothesis based on significance level and p-value

• Not guilty• Guilty

• Collect evidence and present to jury

• Weigh evidence

• Decide if guilty or not guilty

• How do we interpret significance level

• Common sig. Levels

• Have we proven H0 is true or false?

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