Significance Tests Section

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Today’s Objectives Form a null hypothesis and an alternative hypothesis about a population parameter. Find the P-value in support of the alternative hypothesis. Write a conclusion about the evidence in a three-phrase form. California Standard 18.0 Students determine the P- value for a statistic for a simple random sample from a normal distribution.

The Reasoning of a Significance Test Confidence intervals are used to estimate the value of some population parameter. Significance tests are used to support some claim about a parameter. The logic involves these four steps: 1.State what is known about the parameter. a)What value do we ASSUME true? b)What value is CLAIMED true? 2.Gather evidence about the population (such as a sample mean or sample proportion). 3.Ask whether the results found could have happened by chance alone if the assumption (1a) is true. 4.Draw a conclusion about the claim (1b).

Step 1: State What is Assumed and Claimed State the assumption about the population that is held before we gather evidence. –This is called the null hypothesis. –The notation used is H o State the claim that is to be proven. –This is called the alternative hypothesis. –The notation used is H a Write H o and H a as a pair of inequalities. –We normally form H a first because it is easiest to understand. –When actually written on paper, H o comes first.

Example In a criminal trial, the defendant is assumed innocent until proven guilty. The assumption (that he is innocent) is the null hypothesis. The alternative hypothesis is that he is guilty and must be supported by evidence. So the notation would read: H o : Defendant is innocent. H a : Defendant is guilty. Note that we never prove innocence. –We just decide whether there is sufficient evidence to support H a

Step 2: Gather evidence about the population Determine methodology and sample size Collect data Calculate sample mean or sample proportion –This is known as the test statistic

Step 3: Ask whether the evidence gathered could have happened by chance alone Based on the assumption in H o, find the sampling distribution of the statistic. –What are the mean and standard deviation for x-bar or p-hat? Find the probability of getting a statistic as extreme (or more extreme) as the one gathered. –Notice that this probability is based on the assumption that H o is true. –The probability is called the P-value.

Step 4: Draw a conclusion about the claim Is the probability low enough that you believe the statistic gathered could not have happened by chance? –Generally we tend to believe that a 5% or 10% (or higher) event could occur by chance. –If the probability is less than 5%, we tend to get skeptical. (Remember the cards) If the probability is too low, take that as evidence in support of H a

Example I used to own a Baskin-Robbins ice cream store in Pleasant Hill (really!). I trained my employees to make their scoops weigh 3.5 oz. –Specifically, scoops are N(3.5 oz, 0.1 oz) Two of my first employees were Gina Z and Jim Wrenn. –(Yes, the Mr. Wrenn who taught at NHS). My wife thought that Gina and Jim were scooping too much ice cream, so I did a study. I weighed 10 scoops randomly from each. –Jim’s averaged 3.58 oz –Gina’s averaged 3.52 oz Is this evidence to support my wife’s claim, or could this happen by random chance?

Form the hypotheses We are trying to prove that the scoops weigh more than 3.5 oz, so that is the alternative hypothesis. We assume that the scoops are 3.5 oz (or less), so that is the null hypothesis. Here is the notation: H o : µ = 3.5 oz H a : µ > 3.5 oz

Gather evidence and find P-value for Jim Jim’s sample mean was 3.58 Distribution of sample means is N(3.5,.03) –We are assuming H o true –Standard deviation was found by σ/√n formula What is the probability of getting a result as high as 3.58 or higher under this distribution? –normalcdf(3.58, 999, 3.5,.03) =.004 –Note that this is the P-value

Draw a conclusion This says that if Jim’s usual scooping is really N(3.5,.1), there is about a 0.4% probability of getting a sample mean this high (or higher) by chance alone. I don’t believe that an event with such low probability could have happened by chance, so I take this as evidence that the mean of his distribution is really higher than 3.5. –In other words, this is strong evidence to support H a

Write the conclusion You must write your conclusion as a full sentence in context. Use a three-phrase model: –Because the P-value of ### is so (low/high)… –there (is/is not) good evidence to support the claim that… –Re-state the claim. (IN CONTEXT!!!) Follow this model to write a conclusion to this problem now. Because the P-value of 0.4% is so low, there is good evidence to support the claim that Jim’s scoops average more than 3.5 oz.

Do the analysis for Gina Calculate her P-value (based on  = 3.52). Decide whether the result could have happened by chance. Write a three-phrase conclusion. P-value= normalcdf(3.52, 999, 3.5,.03) =.252 A 25% probability could easily happen by chance. Because the P-value of.252 is so high, there is not good evidence to support the claim that Gina’s scoops average more than 3.5 oz.

Three forms of hypotheses If the claim is that the population mean is greater than some value k: –H o : µ=kH a : µ>k If the claim is that the population mean is less than some value k: –H o : µ=kH a : µ<k If the claim is that the population mean is different than some value k: –H o : µ=kH a : µ≠k Note that you must choose H o and H a before gathering data

Today’s Objectives Form a null hypothesis and an alternative hypothesis about a population parameter. Find the P-value in support of the alternative hypothesis. Write a conclusion about the evidence in a three-phrase form. California Standard 18.0 Students determine the P- value for a statistic for a simple random sample from a normal distribution.

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