Chapter 11: Testing a Claim Confidence intervals are one of the two most common types of statistical inference. We use confidence intervals when we wish to estimate a population parameter. The second type of inference is called significance tests. The goal here is to assess the evidence provided by data about some claim concerning the population. Example: The mean number of people per household for ALL United States residents is 2.64. We could collect data on this and compare it to this population mean that is claimed.

Steps of Significance Testing Hypothesis about a population parameter (m or p) Check conditions for testing Collect data to find a test statistic Find a probability that extreme values are possible (p-value) Look at the significance level and interpret the results of the claim The reasoning is that we are looking at what will happen in the long run with repeated sampling or experiments. Significant = not likely to happen by chance

Example: Rosie is an aging sheep dog. Owner is concerned about her heart rate. The veterinarian found that average heart rate for this breed (population) of dog follows a normal distribution with a mean of 115 beats per minute and a standard deviation of 12 beats. A series of heart rates for Rosie have been taken over the last 6 regular check ups.

3 possible hypotheses: Stating the Hypothesis Null Hypothesis (H0): the statement being tested in a significance test (“no effect”, “no difference”, “no change”, “same as the claimed” We are trying to find evidence AGAINST this claim Alternative Hypothesis (Ha): the claim about the population that we are trying to find evidence FOR. These are both stated BEFORE the data (evidence) is provided. 3 possible hypotheses: 𝐻 0 𝜇=115, 𝐻 𝑎 𝜇<115 (if believed to be less than claim) 𝐻 0 𝜇=115, 𝐻 𝑎 𝜇>115 (if believed to be more than claim) 𝐻 0 𝜇=115, 𝐻 𝑎 𝜇≠115 (if just different than claim) * In this example, the vet thinks Rosie’s heart rate is slowing. *

Conditions for Significance Testing SRS Normality For means: population distribution is Normal or n ≥ 30 (CLT) For proportions: 3) Independence As with any type of inference, you MUST check the conditions and state the reasoning for each condition EVERY TIME!!

Test Statistic Collect data and calculate or and standard error We want to compare the sample’s calculations with H0, so we compute a test statistic (z-score) test statistic (z) = estimate ( or ) – H0 value standard error For Rosie… her average heart rate for the 6 check ups is 105. Is this far enough from the claimed mean to consider invalid?

But how large is large enough to prove H0 untrue? P-value: the probability of getting another sample statistic as or more extreme than the current estimate when H0 is assumed true …the probability that the current estimate is a “fluke” The smaller the P-value, the stronger evidence AGAINST H0 To find the P-value, look up the test statistic (z or t) in the table Or 𝒏𝒐𝒓𝒎𝒄𝒅𝒇( −𝟗𝟗𝟗, 𝒙 , 𝑯 𝟎 , 𝒔𝒕𝒅 𝒆𝒓𝒓𝒐𝒓) for less than 𝒏𝒐𝒓𝒎𝒄𝒅𝒇( 𝒙 , 𝟗𝟗𝟗,𝑯 𝟎 , 𝒔𝒕𝒅 𝒆𝒓𝒓𝒐𝒓) for more than For ≠, find P for the one side it’s actually on and DOUBLE For Rosie: P = 0.0206 {𝑛𝑜𝑟𝑚𝑐𝑑𝑓(0, 105, 115, 12 6 )} This means there is about a 2% probability that Rosie’s results happened by chance.

= the max P-value acceptable to give evidence AGAINST H0 To determine if the P-value is “good enough”, we compare that probability to the a significance level (a). = the max P-value acceptable to give evidence AGAINST H0 Significance levels, like confidence levels, are chosen at the discretion of the statistician depending on the situation. typically is set at 0.01, 0.05, or 0.10 We want P ≤ a Example for a = 0.05 Rosie

Interpreting the Results/Draw a Conclusion P ≤ a we reject H0 If H0 is true, the sample results are too unlikely to occur by chance P > a we fail to reject H0 The sample results could possibly occur by chance This does not mean we accept H0 just that there is not enough evidence to reject H0 This does not mean we accept Ha just that H0 is invalid

You try: The Environmental Protection Agency (EPA) has been studying Miller’s Creek regarding the amount of ammonia nitrogen concentration, which can affect plant and animal life in the area. For years, the concentration was 2.3 mg/l. Recently, a new golf course and a house development near the creek have caused concern. A random sample of 8 water tests from the creek were taken. If we assume the concentration levels are normally distributed with a s = 0.30, is there evidence that the ammonia nitrogen concentration levels have changed? (use a = 0.05)