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Tests of Significance about Percents Reading Handout # 6

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Recall Example 1: SAT scores are falling? u 3 years ago -- National AVG = 955 u Random Sample of 200 graduating high school students this year : AVG = 935 SD = 100 Question: Have SAT scores dropped ? Procedure: Determine how “extreme” or “rare” our sample AVG of 935 is if population AVG really is 955.

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u We found the P-value to be.22%

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We must decide: u The sample came from population with population AVG = 955 and just by chance the sample AVG is “small.” OR u We are not willing to believe that the pop. AVG this year is really 955. (Conclude SAT scores have fallen.)

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Hypothesis Testing Logic u Null u Null -- “nothing new is happening” u Alternative u Alternative -- what we “want” to show u Collect data u If - data supports alternative, meaning that “outcomes this extreme in support of the alternative could occur very rarely (< 5% of the time) when null is true” reject the null Then we reject the null.

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SAT Example u We found the P-value to be.22% u So - results were highly statistically significant - i.e. we reject the null hypothesis that AVG is still 955 - we believe SAT scores have fallen

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Example 2 - Last Time (school district data): - P-value = 11.5% - not rare enough - do not reject null - doesn’t mean we have proved AVG=100 (we simply do not have enough evidence to reject it.)

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SO: u If we have very strong evidence against the null, we reject it we believe the alternative is true u If we don’t have very strong evidence against the null, we do not reject it we realize that the null might not be true we do not claim the null is true

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Analogy with Criminal Trial Defendant innocent u Null hypothesis: Defendant innocent Defendant guilty u Alternative hypothesis: Defendant guilty u Decision procedure: Evidence must show guilt beyond a reasonable doubt (before we find guilty) - want only a very small chance that jury finds guilty when person is actually innocent - so there may be fairly large chance jury finds not guilty when defendant is guilty innocent - thus, a finding of not guilty does not necessarily indicate jury believes defendant is innocent

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New Drug - designed to reduce chance of catching common cold Without Drug - chance of catching cold during 1 year period = 40% To Test Drug: 150 subjects selected at random and given drug 35% of these catch cold during next 12 months Is there significant evidence that the drug is effective?

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Suppose that for companies involved with Internet sales, the industry standard is that 90% of orders are mailed within 48 hours. - because of complaints from customers, a consumer group believes a particular company does not meet this standard Consumer Group conducted a survey - 180 orders selected randomly - 157 of these were mailed within 48 hours Is there significant evidence to support this belief?

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