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Statistical Significance and (alpha) level. Lesson 10.2.2. Starter 10.2.2. Write the definition of an unbiased statistic. Today’s Objectives. Determine the significance level ( ) to be used in a hypothesis test
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Statistical Significanceand (alpha) level Lesson 10.2.2
Starter 10.2.2 • Write the definition of an unbiased statistic.
Today’s Objectives • Determine the significance level () to be used in a hypothesis test • Determine whether the results of a hypothesis test are statistically significant • Incorporate P-value and into the conclusion statement of a hypothesis test California Standard 18.0 Students determine the P- value for a statistic for a simple random sample from a normal distribution.
Significance level • If the P-value of a hypothesis test is above 20% or 25%, we would easily agree that there is not enough evidence to reject Ho • If the P-value is less than 20%, we begin to believe that there is some evidence to reject Ho and support Ha • If the P-value is in the range of 10% or 5% or less, we believe that there is good evidence to reject Ho and support Ha • So at what P-value do we decide there is enough evidence? • We resolve this by determining in advance the level of evidence we require to reject Ho • This is called the significance level and denoted by the Greek letter • We decide the level before gathering evidence and state it along with the hypotheses • Then if the P-value is less than the results of a hypothesis test are UNLIKELY TO HAPPEN BY CHANCE and considered STATISTICALLY SIGNIFICANT.
Example • Refer back to the ice cream scooping example • If we define = .10, are the results statistically significant • For Gina? • For Jim? • In Gina’s case, the P-value was about 25% • This is not less than so it is not statistically significant • We fail to reject Ho (or: We continue to assume Ho true) • In Jim’s case, the P-value was about 0.4% • Because this is less than it is statistically significant • we reject Ho and say there is good evidence to support Ha
We need to modify the conclusion statement • The three-phrase conclusion statement for hypothesis tests needs to incorporate a comparison between P and • Instead of saying “Because P is so (low/high)…” say “Because (P < / P > )…” • Continue with the same two phrases as before: • “There (is / is not) good evidence to support the claim that…” • Restate Ha in context • Notice that we still do not “accept” or “prove” Ho • It was assumed true from the outset of the test • If any statement is made about Ho, it would be only that we reject (or fail to reject) the null hypothesis
Example Concluded • Here’s the new conclusion statement for Gina: • Because P (25%) > (10%), there is not good evidence to support the claim that Gina’s mean scoop weight is more than 3.5 oz • Here’s the new conclusion statement for Jim: • Because P (0.4%) < (10%), there is good evidence to support the claim that Jim’s mean scoop weight is more than 3.5 oz
New Example • Suppose we know that the SAT math test nationally follows the N(515, 100) distribution • Mrs. Riley claims that Northgate students score significantly better than the national average • She supports the claim with a study of 100 randomly chosen Northgate students whose average score was 535 • Clearly these results are above the national average. Is this just random chance based on the 100 scores she happened to choose, or is it statistically significant? • Perform a significance test to support the claim • Decide for yourself what significance level is appropriate and write it along with your hypotheses • Do the math (What is the distribution of x-bar? What is P?) • Write a conclusion sentence following the three-phrase form
State hypotheses and significance level • Ho: µ = 515 • Ha: µ > 515 • = .10 or .05 or .01 (depends on your choice) • Gather evidence • We chose 100 random students • The sample mean was: = 535 • Calculate the P-value • The distribution of is N(515, 10) if Ho is true • Then the probability that one particular is 535 or greater is found by the command: normalcdf(535, 999, 515, 10) = .023 • Write the conclusion • If you chose = .01: Since P(2.3%) > (1%) there is not enough evidence to support the claim that the mean Northgate SAT math score is higher than the national average of 515 • If you chose = .10 OR .05: Since P(2.3%) < (5%) there is good evidence to support the claim that the mean Northgate SAT math score is higher than the national average of 515
So what is the true NHS mean? • Use the calculator’s Stat:Test:ZInterval screen to form a 90% confidence interval for the true mean • Remember that σ = 100, Ë = 535, n = 100 • You should have found (518.55, 551.45) • Write a sentence that summarizes this finding • I am 90% confident that the true mean score of Northgate students is between 518.55 and 551.45
Today’s Objectives • Determine the significance level () to be used in a hypothesis test • Determine whether the results of a hypothesis test are statistically significant • Incorporate P-value and into the conclusion statement of a hypothesis test California Standard 18.0 Students determine the P- value for a statistic for a simple random sample from a normal distribution.
Homework • Read pages 540 – 542 • Do problems 33, 35 – 38