P-VALUE ON INFERENTIAL STATISTICS

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Rule of p-value:

If the p-value is small, then we reject the null

The p-value is NOT…

The p-value is not the probability that H0 is true.

A small p-value does not mean H0 is false.

Statistical significanceSometimes, a threshold of evidence called a “significance level” (denoted by α) is set prior to conducting the test.Typically α=0.05, but sometimes people also use α=0.10 , α=0.01 or α=0.001.

If the p-value is less than or equal to α we reject H0. greater than α, we do not reject H0.

A good analogy for hypothesis testing is in the American judicial system: In a trial, there are two competing hypotheses: the defendant is guilty or innocent. Also, the defendant is presumed innocent until they are proven guilty.

For example, we would initially believe that Frito-Lay is telling the truth until we have convincing evidence that the average weight is < 14 oz.

Hypothesis testing

NULL HYPOTHESIS: we initially assume that one of the hypotheses is true

ALTERNATIVE HYPOTHESIS: We then consider the evidence from the sample and reject the null hypothesis (in favor of the)

Significance TestsCall the paramedics!

Vehicle accidents can result in serious injuries to drivers and passengers. When they do, someone calls 911 and paramedic response. Several cities have begun to monitor paramedic response time and they got a = 6.7 minutes with σ=2 minutes. ℳAt the end of the following year, the city manager selects simple random sample of 400 calls and the mean response was x=6.48 minutes. Do these data provide good evidence that response times have decreased since last year?

Stating the hypothesisWe are seeking evidence of decrease in response time this year, thus our null hypothesis says “no decrease” on the average in the large population of all calls involving life-threatening injury this year. The alternative hypothesis says “there is a decrease.” therefore:

Ho: = 6.7 minutesℳHa: < 6.7 minutesℳ

Conditions for significance testingSRS: the city manager took simple random sample of 400 calls for life-threatening injury this year.

Normality: the population distribution may not follow Normal distribution but the sample size of 400 is large enough to ensure Normality of x (by CLT)

Independence: Since the city manager is sampling calls without replacement, we must assume that there were at least (400)(10)=4000 calls involved life-threatening injuries in the city this year.

Test Statistics

test statistic = estimate - hypothesized value

standard deviation of the estimate

ℳ=6.7 minutes x= 6.48 minutesσ= 2 minutesn=400

z= x-ℳσ/√n

z= 6.48-6.72/√400

z= -2.20

P-valuesThe Probability, computed assuming that Ho is true, that the observed outcome would take a

values as extreme as or more extreme that that actually observed is called the P-value of the test.

The smaller the P-value is the stronger the evidence against Ho provided by the data

z= -2.20P=0.0139

ℳ=6.7x=6.48z=-2.20

P=0.0139

Conclusion

ℳ=6.7x=6.48z=-2.20

P=0.0139

There is about 1.4% chance that the city manager would obtain a sample of 400 calls with a mean

response of 6.48 minutes or less. The small P-value provides strong evidence against Ho and in favor the

Ha where <6.7ℳ

Example 2Coffee sales Weekly sales of regular ground coffee at a supermarket have in the recent past varied according to a Normal distribution with mean μ = 354 units per week and standard deviation σ = 33 units. The store reduces the price by 5%. Sales in the next three weeks are 405, 378, and 411 units. Is this good evidence that average sales are now higher? The hypotheses are

H0: μ = 354

Ha: μ > 354

Assume that the standard deviation of the population of weekly sales remains σ = 33.

Solution:

X-bar: 398 µ: 354n:3∂: 33

z= x-ℳσ/√n

z= 398-35433/√3

z= 2.31

p= 0.0104

State the 3 conditions

The small P-value provides strong evidence against

Ho and in favor the Ha where > ℳ

354 units. We can say that there is a pretty convincing evidence that the mean sales are

higher.

Classwork

At the bakery where you work, loaves of bread are supposed to weigh 1 pound. From experience, the weights of loaves produced at the bakery follow a Normal distribution with standard deviation σ = 0.13 pounds. You believe that new personnel are producing loaves that are heavier than 1 pound. As supervisor of Quality Control, you want to test your claim at the 5% significance level. You weigh 20 loaves and obtain a mean weight of 1.05 pounds.

AnswerHo: Loaves of bread has an average weight of µ=1 lb.

Ha: Loaves of bread has an average weight of µ1 lb.Since ∂ is given - use one-sample z-test

SRS: we are not told that the sample is randomly chosen so we should proceed with caution

Normality: the distribution is said to be Normal

Independence: N ≥ 10(20)

Test-statistic:Z=1.72, p=.0427With a p-value of .04247 which is less than .05 significant

level, we have strong evidence to reject the null hypothesis. Therefore, the true average weight of the

loaves of bread is greater than a pound. However, we will proceed with caution since we can’t confirm Normality.

Warm-upStatistics can help decide the authorship of literary works. Sonnets by an Elizabethan poet are known to contain an average of µ= 6.9 new words (words not used in the poet’s other works). The distribution of new words in this poet’s sonnets is Normal with standard deviation ∂= 2.7. Now a manuscript with five new sonnets has come to light, and scholars are debating whether it is the poet’s work. The new sonnets contain an average of x= 9.2 words not used in the poet’s known works. We expect poems by another author to contain more new words than found in the Elizabethan poet’s poems.

Car Pollution