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Tests of Significance 2/4 & 2/5. Chapter 10. 10.10&12. Warm Up. Review your homework, ONLY #1,6,13, and see if you have an questions. . Questions about #1,6, & 13. Use a confidence interval when your goal is to estimate a population parameter

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tests of significance

Use a confidence interval when your goal is to estimate a population parameter

Use a Test of Significance to assess the evidence provided by data about some claim concerning a population.

Tests of Significance
tests of significance1

I make 80% of my free throws in basketball. You don’t believe me and want to test my claim. You make me shoot 20 free throws and I only make 8 out of 20.

  • Do you still believe my claim?
    • No. Your reasoning is based on asking what would happing if my claim were true and we repeated the sample of 20 free throws many times. In fact, there is only a .0001% probability that I would only make 8 shots out of 20 if I really did make 80% of all my shots. This would only happen once out of 10,000 tries in my claim were true. This small probability convinces you that I am wrong.
Tests of Significance
how a significance test works

The null hypothesis is the statement being tested in a test of significance.

    • The null hypothesis is normally the statement of “no effect” or “no difference”
  • Works by asking how unlikely the observed outcome would be if the null hypothesis was true.
    • The less likely the outcome, the stronger the evidence against
  • We measure the strength of the evidence against by the probability under the normal curve to the right of the observed .
    • The probability is called the P-value.
    • A common rule of thumb – a p-value less than 0.05 is called statistically significant.
How a significance test works
outline of the test

Describe the effect you are searching for in terms of a population parameter

    • State the null hypothesis and the alternative hypothesis.
  • From the data, calculate a statistic like that estimates the parameter.
    • Is this statistic far from the parameter value stated in the null hypothesis?
  • Find the p-value.
    • A small p-value indicates that the statistic would be unlikely if the null hypothesis was true.
    • Therefore, a small p-value gives evidence against the null hypothesis
Outline of the test
inference toolbox significance tests

To test a claim about an unknown population parameter:

Step 1: Identify the population of interest and the parameter you want to draw conclusions about. State the null and alternative hypotheses in words and symbols.

Step 2: Choose the appropriate inference procedure. Verify the conditions for using the selected procedure.

SRS from population of interest

Sampling distribution is approx. normal.

Step 3: If the conditions are met, carry out the inference procedure.

Calculate the test statistic Find the P-value

Step 4: Interpret your results in the context of the problem.

Inference Toolbox: Significance Tests
sweetening colas

Diet colas use artificial sweeteners, and these sweeteners lose their sweetness over time. The manufacturer tests the sweetness of a diet cola after simulating four months of storage at room temperature. The trained testers rate the diet cola on a sweetness score from 1 – 10.

We want to know if the colas really do lose their sweetness.

Let’s walk through the steps of a test of significance together.

Sweetening Colas
sweetening colas1

Step 1:

    • The population of interest is all diet colas
    • The parameter we wish to infer about is the mean loss of sweetness in colas
    • The null hypothesis is that the colas lost no sweetness and the alternative hypothesis is that the colas did lose sweetness.
  • Step 2:
    • We are to use a test of significance.
    • The sample is an SRS from the population of interest.
    • The sampling distribution is approximately normal because the population is normal.
Sweetening Colas
sweetening colas continued

Step 3:

    • Since the conditions are met, we will carry out the inference procedure.
    • The test statistic is computed by the one-sample z statistic:
      • =
    • To find the P-value we use Table A. Since our alternative hypothesis is one-sided and it is a greater than hypothesis, we look at the area to the right of the z-score.
      • P-value: .000628
Sweetening Colas, continued.
sweetening colas continued1

Step 4:

    • Since our P-value is very small, it is very unlikely that we would find a mean loss of sweetness of 1.02 in a sample of ten colas. We say that we reject the null hypothesis in favor of the alternative hypothesis that the mean loss of sweetness is greater than zero.
Sweetening Colas, continued.
alternative hypothesis

 one-sided

    • We are only interested in deviations from the null hypothesis in one direction
  •  two-sided
    • When a direction is not specified in the problem.
    • For example, if a manager wants to know if there is a difference between the pace of machines vs. the pace of human workers
      • &
Alternative Hypothesis
p values

The decisive value of P is called the significance level, denoted .

    • If we choose =0.05, we are requiring that the data give evidence against so strong that it would happen no more than 5% of the time when is true.
    • What if =0.01?
  • If , we say that the data are statistically significant at level
P-values
calculating a one sided p value

Cola example.

  • Suppose that one cola had .
  • Hence, the P-value for test the hypotheses is ), assuming that is true.
    • Hence, has normal distribution with mean 0 and standard deviation given by 0.316.
  • Standardize :
Calculating a one-sided P-value
one sided p value

Hence, if is true, and the mean sweetness loss for this cola is 0, there is about a 17% chance that we will obtain a sample of 10 sweetness loss values whose mean is 0.3 or greater.

This can happen by chance, easily.

Not strong evidence against the null hypothesis.

One-sided P-value
two sided p value

If our alternative hypothesis is two-sided we will find a two-sided P-value.

  • Since the standard normal curve is symmetric, we can calculate this probability by doubling it.
Two Sided P-Value
two sided p value example

10.38

The target value for the hardness of the pill . The hardness data for a sample of 20 pills are:

Is there significant evidence at the 5% level that the mean hardness of the pills is different from the target value? Use the Inference Toolbox

Two Sided P-value Example
slide22

Step 1: The population of interest is the pills and the parameter we want to draw conclusions about is the mean hardness. The null hypothesis is that the mean hardness is equal to 11.5 while the alternative hypothesis is that the mean hardness is not equal to 11.5

  • Step 2: We will use a significance test. The data comes from an SRS from the population of interest. The sampling distribution of is approximately normal because n=20 is large enough by the CLT.
slide23

Step 3: Calculate test statistic:

    • We know that
    • Then, the one-sample test statistic is Find P-value:
slide24

Step 4: Interpret results in context.

    • We fail to reject the null hypothesis because this is reasonable variation when the null hypothesis is true.
    • So, if the population mean really is 11.5, then it would not be abnormal to get a sample of 20 pills with a mean of 11.5164.