A hypothesis is a claim Population mean The mean monthly cell phone bill in this city is ? = $42 Population proportion Example: The proportion of adults in this city with cell phones is ? = 0. 68 States the claim or assertion to be tested Is always about a population parameter, not about a sample statistic Is the opposite of the null hypothesis e. g. , The average diameter of a manufactured bolt is not equal to 30mm ( H1: ? ? 30 ) Challenges the status quo Alternative never contains the “=”sign May or may not be proven
Is generally the hypothesis that the researcher is trying to prove Is the opposite of the null hypothesis e. g. , The average diameter of a manufactured bolt is not equal to 30mm ( H1: ? ? 30 ) Challenges the status quo Alternative never contains the “=”sign May or may not be proven Is generally the hypothesis that the researcher is trying to prove Is the opposite of the null hypothesis e. g. , The average diameter of a manufactured bolt is not equal to 30mm ( H1: ? ? 30 ) Challenges the status quo Alternative never contains the “=”sign May or may not be proven
Is generally the hypothesis that the researcher is trying to prove If the sample mean is close to the stated population mean, the null hypothesis is not rejected. If the sample mean is far from the stated population mean, the null hypothesis is rejected. How far is “far enough” to reject H0? The critical value of a test statistic creates a “line in the sand” for decision making -- it answers the question of how far is far enough. Type I Error Reject a true null hypothesis Considered a serious type of error The probability of a Type I Error is ? Called level of significance of the test
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Set by researcher in advance Type II Error Failure to reject a false null hypothesis The probability of a Type II Error is ? Type I and Type II errors cannot happen at the same time A Type I error can only occur if H0 is true A Type II error can only occur if H0 is false Critical Value Approach to Testing For a two-tail test for the mean, ? known: Determine the critical Z values for a specified level of significance ? from a table or computer Decision Rule: If the test statistic falls in the rejection region, reject H0 ; otherwise do not reject H0
State the null hypothesis, H0 and the alternative hypothesis, H1 Determine the appropriate test statistic and sampling distribution Determine the critical values that divide the rejection and nonrejection regions Collect data and compute the value of the test statistic Make the statistical decision and state the managerial conclusion. If the test statistic falls into the nonrejection region, do not reject the null hypothesis H0. If the test statistic falls into the rejection region, reject the null hypothesis. Express the managerial conclusion in the context of the problem p-Value Approach to Testing -value: Probability of obtaining a test statistic equal to or more extreme than the observed sample value given H0 is true The p-value is also called the observed level of significance H0 can be rejected if the p-value is less than ? Hypothesis Testing: ? Unknown If the population standard deviation is unknown, you instead use the sample standard deviation S. Because of this change, you use the t distribution instead of the Z distribution to test the null hypothesis about the mean. When using the t distribution you must assume the population you are sampling from follows a normal distribution.
All other steps, concepts, and conclusions are the same. One-Tail Tests In many cases, the alternative hypothesis focuses on a particular direction H0: ? ? 3 H1: ? < 3 This is a lower-tail test since the alternative hypothesis is focused on the lower tail below the mean of 3 H0: ? ? 3 H1: ? > 3 This is an upper-tail test since the alternative hypothesis is focused on the upper tail above the mean of 3 Proportions Sample proportion in the category of interest is denoted by p When both X and n – X are at least 5, p can be approximated by a normal distribution with mean and standard deviation
Potential Pitfalls and Ethical Considerations Use randomly collected data to reduce selection biases Do not use human subjects without informed consent Choose the level of significance, ? , and the type of test (one-tail or two-tail) before data collection Do not employ “data snooping” to choose between one-tail and two-tail test, or to determine the level of significance Do not practice “data cleansing” to hide observations that do not support a stated hypothesis Report all pertinent findings including both statistical significance and practical importance
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