s1 w6 hypothesis

• Hypothesis Testing


  A procedure for testing a claim about a property of a population
• Main activities of inferential statistics


  • Estimating a population parameter with a confidence interval
  • Testing a hypothesis or claim about a population parameter
• Null hypothesis (H0)


  A statement that the value of a population parameter (such as proportion, mean, or standard deviation) is equal to some claimed value
• Alternative hypothesis (H1 or HA)


  The statement that the parameter has a value that somehow differs from the null hypothesis
• The symbolic form of the alternative hypothesis must use one of these symbols: <, >, ≠

  
• Rare Event Rule for Inferential Statistics


  If, under a given assumption, the probability of a particular observed event is exceptionally small, we conclude that the assumption is probably not correct
• Forming your own claims (hypotheses)

  The claim must be worded so that it becomes the alternative hypothesis

  
• When conducting hypothesis tests, consider the context of the data, the source of the data, and the sampling method used to obtain the sample data
• Components of a hypothesis test

  • Identifying the null hypothesis and alternative hypothesis from a given claim, and expressing both in symbolic form
  • Calculating the value of the test statistic, given a claim and sample data
  • Choosing the sampling distribution that is relevant
  • Identifying the P-value or identifying the critical value(s)
  • Stating the conclusion about a claim in simple and nontechnical terms

  
• Significance level (α)


  The probability that the test statistic will fall in the critical region when the null hypothesis is actually true (making the mistake of rejecting the null hypothesis when it is true)
• Test statistic


  A value used in making a decision about the null hypothesis, found by converting the sample statistic to a score with the assumption that the null hypothesis is true
• Finding the value of the test statistic and then finding either the P-value or the critical value(s)

  1. First transform the relevant sample statistic to a standardized score called the test statistic

  2. Then find the P-Value or the critical value(s)
• 1. value


  The probability of getting a value of the test statistic that is at least as extreme as the one representing the sample data, assuming that the null hypothesis is true
• Critical region


  The set of all values of the test statistic that cause us to reject the null hypothesis
• Critical value


  Any value that separates the critical region (where we reject the null hypothesis) from the values of the test statistic that do not lead to rejection of the null hypothesis
• Do not confuse a P-value with a population proportion p
• Decision Criterion


  1. value Method: If P-value ≤ α, reject H0. If P-value > α, fail to reject H0.

  Critical Value Method: If the test statistic falls within the critical region, reject H0. If the test statistic does not fall within the critical region, fail to reject H0.

  
• For the XSORT baby gender test, there was not sufficient evidence to support the claim that the XSORT method is effective in increasing the probability that a baby girl will be born
• Never conclude a hypothesis test with a statement of "reject the null hypothesis" or "fail to reject the null hypothesis"
• 1.60 and a P-Value of 0.0548
• We tested: H0 : p = 0.5, H1 : p > 0.5
• Using the P-Value method, we would fail to reject the null at the α = 0.05 level

  
• Using the critical value method, we would fail to reject the null because the test statistic of z = 1.60 does not fall in the rejection region
• We come to the same decision using either method
• We are not proving the null hypothesis, we are failing to reject it
• Fail to reject says more correctly that the available evidence is not strong enough to warrant rejection of the null hypothesis
• Two common methods for testing a claim about a population proportion are: 1) to use a normal distribution as an approximation to the binomial distribution, and 2) to use an exact method based on the binomial probability distribution
• n


  Sample size or number of trials
• p


  Population proportion
• q


  1 - p
• The sample observations are a simple random sample
• The conditions for a binomial distribution are satisfied
• The conditions np ≥ 5 and nq ≥ 5 are both satisfied, so the binomial distribution of sample proportions can be approximated by a normal distribution with μ = np and σ=npq
• p is the assumed proportion not the sample proportion
• 1. value


  Probability of getting a test statistic at least as extreme as the one representing sample data
• Computer programs and calculators usually provide a P-value, so the P-value method is used
• When testing claims about a population proportion, the traditional method and the P-value method are equivalent and will yield the same result since they use the same standard deviation based on the claimed proportion p
• The confidence interval uses an estimated standard deviation based upon the sample proportion p̑
• Consequently, it is possible that the traditional and P-value methods may yield a different conclusion than the confidence interval method
• A good strategy is to use a confidence interval to estimate a population proportion, but use the P-value or traditional method for testing a claim about the proportion