Chapter 15: Tests of Significance: The Basics STAT 1450

Connecting Chapter 15 to our Current Knowledge of Statistics ▸ Chapters 10 & 12 used information about a population to answer questions about a sample. (e.g., 20% of people are smokers, what is the probability that a random sample of 2 people smoke). ▸ Now with inference, we have statistics problems where we use the information about a sample to answer questions concerning the population Tests of Significance

Connecting Chapter 15 to our Current Knowledge of Statistics ▸ If we want to estimate a population parameter, then we should use statistics to create a confidence interval. ▸ If, on the other hand, we want to assess the evidence provided by data about some claim concerning a population parameter, we need to do a hypothesis test Tests of Significance

The Reasoning of Tests of Significance ▸ We are now inquiring about a behavior of an event if a phenomena was repeated numerous times. We will begin by working with simple random samples of data from Normal populations with known standard deviations The Reasoning of Tests of Significance

The Reasoning of Tests of Significance ▸ Situation: People drink coffee for a variety of professional, and now, social reasons. Coffee used to merely be a beverage option on the menu. Now, it is the main attraction for a growing number of restaurants and shoppes. The standard “cup of coffee” is 8 oz. However, even a Tall at Starbuck’s is 12 oz The Reasoning of Tests of Significance

The Reasoning of Tests of Significance Please answer the following questions: a.How many ounces of coffee do you think people typically drink each day? ____ b.How many ounces of coffee do you drink daily? ____ Note: We will only consider the population of regular coffee drinkers The Reasoning of Tests of Significance

Null and Alternative Hypotheses ▸ We have two possible hypotheses about this situation: 1. The mean amount of coffee consumed daily is not different than the value listed in a). 2. The mean amount of coffee consumed daily is different from the value listed in a). ▸ These hypotheses have names:  The null hypothesis (denoted H 0 ) is the claim tested about the population parameter. The test is designed to assess the strength of the evidence against the null hypothesis. Usually the null hypothesis is a statement of “no effect” or “no difference.” It commonly assumes the “benefit of the doubt.”  The alternative hypothesis (denoted H a ) is the claim about the population parameter that we are trying to find evidence for Stating Hypotheses

One- and Two-sided Alternative Hypotheses ▸ An alternative hypothesis is one-sided if it states that a parameter is larger than or smaller than the null hypothesis value. ▸ It is two-sided if it states that the parameter is different from the null value. (It could be either smaller or larger.) ▸ Question: Is the alternative hypothesis in our situation one-sided or two-sided? 15.2 Stating Hypotheses

One- and Two-sided Alternative Hypotheses ▸ An alternative hypothesis is one-sided if it states that a parameter is larger than or smaller than the null hypothesis value. ▸ It is two-sided if it states that the parameter is different from the null value. (It could be either smaller or larger.) ▸ Question: Is the alternative hypothesis in our situation one-sided or two-sided?  Testing a “difference” implies that we are interested in whether the mean daily coffee consumption amount is either greater than, or, less than our estimate Stating Hypotheses

Example: Coffee Consumption ▸ Let’s use one of the values from a) to compose the null and alternative hypotheses Stating Hypotheses

Example: Coffee Consumption 15.2 Stating Hypotheses

Example: Coffee Consumption ▸ Suppose it is known that the standard deviation for daily coffee consumption is 9.2 oz. The average amount of coffee consumed daily for a random sample of 48 people is oz. ▸ True or False: We know with certainty that the average amount of coffee consumed daily is different from our hypothesized value Stating Hypotheses

Example: Coffee Consumption ▸ Suppose it is known that the standard deviation for daily coffee consumption is 9.2 oz. The average amount of coffee consumed daily for a random sample of 48 people is oz. ▸ True or False: We know with certainty that the average amount of coffee consumed daily is different from our hypothesized value. ▸ FALSE. It may be likely given our sample size, but not with certainty. There is variability in sampling. It is possible that the next random sample of 48 could yield a different average Stating Hypotheses

Stating Hyptotheses ▸ Important note: Base your alternative hypothesis on your question of interest—do not base it on the data Stating Hypotheses

Definition: Test Statistic ▸ A test statistic calculated from the sample data measures how far the data departs from what we would expect if the null hypothesis were true. ▸ The further this statistic is from 0, the more the data contradicts the null hypothesis.  Note: A test statistic tells us how many standard deviations our value is away from the hypothesized mean. A positive test statistic is above the mean. A negative one is below the mean. We then use this information to figure out how likely it is to see results like ours if the null hypothesis was true P-values & Statistical Significance

Definition: P-value ▸ The probability, computed assuming that the null hypothesis is true, that the test statistic would take a value as extreme or more extreme than that actually observed is called the P-value (probability value) of the test P-values & Statistical Significance

P-values ▸ If the P-value is small enough, the data we observed would be unusual (very unlikely to have happened) if the null hypothesis were true. ▸ If the P-value is not small enough, the data we observed are not strange at all (could plausibly have happened due to sampling variability) if the null hypothesis were true P-values & Statistical Significance

Tests of Significance & the Justice System 15.3 P-values & Statistical Significance Tests of SignificanceJustice System Null HypothesisThe defendant gets the “benefit of the doubt.” (i.e., they are not guilty). Alternative HypothesisThey are “guilty.” Test StatisticTotality of Evidence collected. P-valueThe probability of observing data as extreme as what was collected under the assumption that the defendant is, indeed, “not guilty.” When the evidence collected seems ‘likely’ (based upon the null hypothesis) DecisionJury rules that the defendant is ‘not guilty.”

Tests of Significance & the Justice System 15.3 P-values & Statistical Significance Tests of SignificanceJustice System When the evidence collected seems ‘likely’ (based upon the null hypothesis) DecisionJury rules that the defendant is ‘not guilty.” When, the evidence collected seems ‘extremely unlikely’ (based upon the null hypothesis) DecisionEither we have “bad” data (mistrial, tampering, etc…) -Or- The jury rules that the defendant is ‘guilty.’

Tests of Significance & the Justice System 15.3 P-values & Statistical Significance ▸ Note: Our jury system assumes innocent until proven guilty. ▸ The actual truth of whether the person indeed committed the crime may never be known.

Significance Level ▸ Question: What is the cut-off between “likely,” “unlikely,” and “extremely unlikely?” ▸ If the P-value is as small or smaller than , we say that the data are ▸ statistically significant at level . The quantity  is called the significance level or the level of significance P-values & Statistical Significance

Significance Level ▸ Question: What is the cut-off between “likely,” “unlikely,” and “extremely unlikely?”  Most common is P-value <  “Likely” prob. >.10“Unlikely” 5% to 10%“Extremely Unlikely” < 5% ▸ If the P-value is as small or smaller than , we say that the data are ▸ statistically significant at level . The quantity  is called the significance level or the level of significance P-values & Statistical Significance

Significance Level ▸ After choosing an appropriate level of significance, we can make a decision about H P-values & Statistical Significance P-Value vs. α Decisions about H 0 P-value > α H o P-value ≤ α H o

Significance Level ▸ After choosing an appropriate level of significance, we can make a decision about H P-values & Statistical Significance P-Value vs. α Decisions about H 0 P-value > α Fail to Reject H o P-value ≤ α Reject H o

Significance Level ▸ Question: Why should a significance level be set before the test has been done?  Suppose that your P-value = If  = 0.05, you would reject the null hypothesis; if  = 0.01, you would fail to reject the null hypothesis. If you did not set a significance level before the test, you might change your mind based on the results to fit the decision you (might) desire. ▸ The test statistic for hypothesis testing has is based upon our work from sampling distributions and confidence intervals P-values & Statistical Significance

Tests for a Population Mean ▸ Tests of significance, allow researchers to determine the validity of certain hypotheses based upon P-values. There are various parameters that we can test (proportions, standard deviations, etc…). ▸ We will begin with the most common parameter to be tested, the mean; much like how we began our confidence interval discussion by estimating the true mean, µ Tests for a Population Mean

Tests for a Population Mean 15.4 Tests for a Population Mean

Tests for a Population Mean 15.4 Tests for a Population Mean Key Words Alternative HypothesisP-value Rejection Region (“extremely unlikely values”)

Tests for a Population Mean 15.4 Tests for a Population Mean Key Words Alternative HypothesisP-value Rejection Region (“extremely unlikely values”) “more than” “increased”

Tests for a Population Mean 15.4 Tests for a Population Mean Key Words Alternative HypothesisP-value Rejection Region (“extremely unlikely values”) “more than” “increased” “less than” “reduced”

Tests for a Population Mean 15.4 Tests for a Population Mean Key Words Alternative HypothesisP-value Rejection Region (“extremely unlikely values”) “more than” “increased” “less than” “reduced” “different” “is not”

Tests for a Population Mean: 4-Step Process 1.State:What is the practical question that requires a statistical test? 2.Plan: a. Identify the parameter b.List all given information from the data collected. c.State the null (H 0 ) and alternative (H A ) hypotheses. d.Specify the level of significance. e.Determine the type of test.(Left-tailed, Right-tailed, Two-Tailed) f.Sketch the region(s) of “extremely unlikely” test statistics. (Rejection Region(s) ) 15.4 Tests for a Population Mean

Tests for a Population Mean: 4-Step Process 3.Solve: a.Check the conditions for the test you plan to use. (Random sample? Large population:sample ratio? Large enough for sample?) b.Calculate the test statistic. c.Determine (or estimate) the P-Value. 4.Conclude: a.Make a decision about the about the null hypothesis. (Reject H 0 or Fail to reject H 0 ). b.Interpret the decision in the context of the original claim. (i.e.,“There is enough (or not enough) evidence at the α level of significance that …) 15.4 Tests for a Population Mean

Example: Coffee Consumption ▸ The standard deviation of daily coffee consumption is 9.2 oz. A random sample of 48 people consumed an average of oz. of coffee daily. Is this evidence that the average amount of coffee consumed daily is different from our original estimate? 15.4 Tests for a Population Mean

Example: Coffee Consumption Is the average amount of coffee consumed daily not 20 oz.? 15.4 Tests for a Population Mean

Example: Coffee Consumption 15.4 Tests for a Population Mean

Example: Coffee Consumption 15.4 Tests for a Population Mean

Example: Coffee Consumption 15.4 Tests for a Population Mean

Example: Coffee Consumption 15.4 Tests for a Population Mean

Example: Coffee Consumption 15.4 Tests for a Population Mean

Example: Coffee Consumption 15.4 Tests for a Population Mean

Example: Coffee Consumption 15.4 Tests for a Population Mean

Example: Coffee Consumption 15.4 Tests for a Population Mean

Example: Coffee Consumption 15.4 Tests for a Population Mean

Example: Coffee Consumption 15.4 Tests for a Population Mean

Technology Tips – Conducting Tests of Significance (σ known) 15.4 Tests for a Population Mean

Technology Tips – Conducting Tests of Significance (σ known) 15.4 Tests for a Population Mean

Technology Tips – Conducting Tests of Significance (σ known) 15.4 Tests for a Population Mean

Example: IQ Scores ▸ Recall that IQ scores from Chapter 14 followed a Normal Distribution with  = 15. You suspect that persons from affluent communities have IQ scores above 100. A random sample of 35 residents of an affluent community had an average IQ score of 112. Is there significant evidence to support your claim at the  =.05 level? 15.4 Tests for a Population Mean

Example: IQ Scores ▸ Recall that IQ scores from Chapter 14 followed a Normal Distribution with  = 15. You suspect that persons from affluent communities have IQ scores above 100. A random sample of 35 residents of an affluent community had an average IQ score of 112. Is there significant evidence to support your claim at the  =.05 level? STATE: Do affluent communities have a mean IQ score above 100? 15.4 Tests for a Population Mean

Example: IQ Scores Plan: a. Identify the parameter  = average IQ score for affluent communities 15.4 Tests for a Population Mean

Example: IQ Scores Plan: a. Identify the parameter  = average IQ score for affluent communities b.List all given information from the data collected._n=35, _  = 150, 15.4 Tests for a Population Mean

Example: IQ Scores Plan: a. Identify the parameter  = average IQ score for affluent communities b.List all given information from the data collected._n=35, _  = 150, c.State the null (H 0 ) and alternative (H A ) hypotheses. H 0 : _  = 100 H A : _  > Tests for a Population Mean

Example: IQ Scores Plan: a. Identify the parameter  = average IQ score for affluent communities b.List all given information from the data collected._n=35, _  = 150, c.State the null (H 0 ) and alternative (H A ) hypotheses. H 0 : _  = 100 H A : _  > 100__ d.Specify the level of significance.  = Tests for a Population Mean

Example: IQ Scores Plan: a. Identify the parameter  = average IQ score for affluent communities b.List all given information from the data collected._n=35, _  = 150, c.State the null (H 0 ) and alternative (H A ) hypotheses. H 0 : _  = 100 H A : _  > 100__ d.Specify the level of significance.  =.05 e. Determine the type of test.(Left-tailed, Right-tailed, Two-Tailed) 15.4 Tests for a Population Mean

Example: IQ Scores Plan: a. Identify the parameter  = average IQ score for affluent communities b.List all given information from the data collected._n=35, _  = 150, c.State the null (H 0 ) and alternative (H A ) hypotheses. H 0 : _  = 100 H A : _  > 100__ d.Specify the level of significance.  =.05 e. Determine the type of test.(Left-tailed, Right-tailed, Two-Tailed) f.Sketch the region(s) of “extremely unlikely” test statistics. (Rejection Region(s) ) 15.4 Tests for a Population Mean

Tests for a Population Mean: 4-Step Process 3.Solve: a.Check the conditions for the test you plan to use. Random sample? Large population : sample ratio? Large enough for sample? Yes Certainly more than 35 We were informed that affluent households the data came from a Normal Distribution Tests for a Population Mean

Tests for a Population Mean: 4-Step Process 3.Solve: a.Check the conditions for the test you plan to use. Random sample? Large population : sample ratio? Large enough for sample? Yes Certainly more than 35 We were informed that affluent households the data came from a Normal Distribution. b.Calculate the test statistic Tests for a Population Mean

Tests for a Population Mean: 4-Step Process 3.Solve: a.Check the conditions for the test you plan to use. Random sample? Large population : sample ratio? Large enough for sample? Yes Certainly more than 35 We were informed that affluent households the data came from a Normal Distribution. b.Calculate the test statistic. c. Determine (or estimate) the P-Value P(Z > 4.73) = Tests for a Population Mean

Tests for a Population Mean: 4-Step Process 4. Conclude: a.Make a decision about the null hypothesis. P-value =.0001 ≤.05 =  (default) (Reject H 0 or Fail to reject H 0 ) Tests for a Population Mean

Tests for a Population Mean: 4-Step Process 4. Conclude: a.Make a decision about the null hypothesis. P-value =.0001 ≤.05 =  (default) (Reject H 0 or Fail to reject H 0 ). b.Interpret the decision in the context of the original claim. (i.e.,“There is enough (or not enough) evidence at the α level of significance that …) 15.4 Tests for a Population Mean

Tests for a Population Mean: 4-Step Process 4. Conclude: a.Make a decision about the null hypothesis. P-value =.0001 ≤.05 =  (default). (Reject H 0 or Fail to reject H 0 ). b.Interpret the decision in the context of the original claim. (i.e.,“There is enough (or not enough) evidence at the α level of significance that …)  There is enough evidence at the  =.05 level to conclude that persons from affluent communities have a mean IQ score above Tests for a Population Mean

Technology Tips – Conducting Tests of Significance (σ known) 15.4 Tests for a Population Mean

Technology Tips – Conducting Tests of Significance (σ known) 15.4 Tests for a Population Mean

Final Note ▸ Some homework exercises will provide you with raw data. ▸ You are to use the data to compute the sample mean and/or standard deviation. ▸ Then proceed with computing the confidence interval or performing a test of significance Tests for a Population Mean

Final Note 15.4 Tests for a Population Mean Steps for Success- Conducting Tests of Significance 1.Set up your Hypotheses.

Final Note 15.4 Tests for a Population Mean Steps for Success- Conducting Tests of Significance 1.Set up your Hypotheses. 2.Check your Conditions.

Final Note 15.4 Tests for a Population Mean Steps for Success- Conducting Tests of Significance 1.Set up your Hypotheses. 2.Check your Conditions. 3. Compute the Test Statistic.

Final Note 15.4 Tests for a Population Mean Steps for Success- Conducting Tests of Significance 1.Set up your Hypotheses. 2.Check your Conditions. 3. Compute the Test Statistic. 4. Compute the P-Value.

Final Note 15.4 Tests for a Population Mean Steps for Success- Conducting Tests of Significance 1.Set up your Hypotheses. 2.Check your Conditions. 3. Compute the Test Statistic. 4. Compute the P-Value. 5. Make a Decision.

Significance from a Table ▸ The graphing calculator and JMP provide the most accurate P-value calculations. Tables can also be used to estimate P-values. ▸ There are two methods of determining the P-Value for a z-statistic Significance from a Table

Significance from a Table (Method 1—Table C) This method uses Table C (page 679 in the textbook). 1.Compare z with the critical values z* and the bottom of Table C. 2.If Z falls between two values of z*, then the P-value falls between the two corresponding values of P in the “One-sided P” or the “Two-sided P” row of Table C Significance from a Table

Significance from a Table (Method 2—Table A) This method uses Table A (page 676 in the textbook). 1.Compute the P-value, which is: a.P (Z > z) for a Right-tailed test. b.P (Z < z) for a Left-tailed test. c.2*P (Z > |z|) for a two-tailed test. 2.Compare the P-value with α Significance from a Table

Significance from a Table ▸ Using technology to compute P-values is most preferred. ▸ For our purposes, using Table C is a suitable alternative to technology. This may not produce the same accuracy as the other options, but it will strengthen estimation skills Significance from a Table

Example: P(Z< -1.45) via Table C ▸ We need to estimate P (Z < -1.45) via Table C. 1.Search the z* row of Table C is between and Significance from a Table

Example: P(Z< -1.45) via Table C ▸ We need to estimate P (Z < -1.45) via Table C. 1.Search the z* row of Table C is between and Therefore its P-value is between.10 and.05 (similar to a 1-sided test) < P-value < Significance from a Table

Example: P(Z< -1.45) via Table A ▸ We need to estimate P (Z < -1.45) via Table A. 1.Compute P( Z < -1.45) from Table A Significance from a Table

Example: P(Z< -1.45) via Table A ▸ We need to estimate P (Z < -1.45) via Table A. 1.Compute P( Z < -1.45) from Table A. 2.P(Z < -1.45) = ▸ Notice that this coincides with the answer from Table C. ▸ Table C will allow us to work with distributions where the population standard deviation is unknown (this will be covered in Chapter 18) Significance from a Table

Five-Minute Summary ▸ List at least 3 concepts that had the most impact on your knowledge of tests of significance. _______________________________________