AP Statistics: Unit 8

Significant Tests and Inferencing:

• Types of inferencing: use a confidence interval to estimate a population parameter and use a significance test to assess the evidence provided by data concerning a parameter

• A significance test is a formal procedure for using observed data to decide between two competing claims (also called hypotheses). The claims are often statements about a parameter, such as the population proportion p or the population mean 𝝁

Interpreting a Hypothesis:

• The alternative hypothesis is one-sided if it states that a parameter is larger than the null hypothesis value or that the parameter is smaller than the null value. 

• It is two-sided if it states that the parameter differs from the null hypothesis value (it could be either larger or smaller). 

• Null Hypothesis: H0

• Alternative Hypothesis: Ha

• Small P-values are evidence against H0 because they say that the observed result is unlikely to occur when H0 is true. (significant result)

• Large P-values fail to give convincing evidence against H0 and in favor of Ha because they say that the observed result is likely to occur by chance alone when H0 is true. (not a significant result)

Interpretation Examples:

• Problem:

  (a)  Explain what it would mean for the null hypothesis to be true in this setting

  Mean=1300 says the mean daily calcium intake in the population of teenagers is 1300mg. If this is true, it means the teenagers are getting enough calcium. 

  (b)  Interpret the P-value in context. 

  Assuming that the mean daily calcium consumption in the population of teens is 1300mg, there is a 0.1404 probability of getting a sample mean of 1198mg or less just by chance in a random sample of 20 teens.

• Problem: H0: mu = 30 hours Ha: mu > 30 hours

  where mu is the true mean lifetime of the new deluxe AAA batteries. The resulting P-value is 0.0729.

  Problem: What conclusion would you make for each of the following significance levels? Justify your answer.

  (a) alpha=0.10 (b) alpha=0.05

  1. Due to the p-value of 0.0729 being smaller than the level of 0.10. We reject the null hypothesis because we have convincing evidence that the company’s deluxe AA batteries last longer than 30 hours on average. 

  2. Due to the p-value 0.0729 being greater than the level of 0.05, we fail to reject the null hypothesis and we have no convincing evidence that the company deluxe AA batteries last longer than 30 hours. 

Checking Conditions Before Significant Testing and Interpretation:

• Check a random sampling method is being used

• Check for independent trials (if no replacement check 10% condition)

• Check normal distribution using the central limit theorem

• Check the large counts rule (nxpo>10 and nx(1-po)>10) or CLT

  
  

Calculations: Test Statistic and P-value:

• Standard Deviation Formula: √p0(1-p0)/n

• Test Statistic Formula: Test statistic = (statistic - parameter) / standard deviation of statistic

  

• Using Z (test statistic) enter normalcdf to get the p-value

• Alternate formula: using technology (1-PropZ-Test) or (T-Test)

• Formula for mean test statistic:

  

T-Test vs. 1-PropZ-Test

• T-Test = problems that deal with means/averages

• 1-PropZ-Test = problems that deal with proportions/percents

Type 1 and Type II Errors:

• Type 1 = occurs when you reject a null hypothesis that is actually true (a "false positive")

• Type II = happens when you fail to reject a null hypothesis that is actually false (a "false negative")