Statistics Note Sheet

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Practice Question #1: A gambler claims he can predict the roll of a die more often than chance would predict. To test this, a die is rolled 100 times and the gambler guesses the number that comes up twenty times and guesses incorrectly the other eighty times. Is this strong evidence that the gambler’s claim is true? a. Specify the null and alternative hypothesis for this problem. p = probability gambler guess correctly for an individual roll null (Ho): p = 1/6 alternative (Ha): p > 1/6

b. Find the test statistic and calculate the p-value. What do you conclude? pˆ = 0.2 so z = (0.2-0.1667)/ .1667(.8333) /100 = 0.894. From the table the p-value is between 18% and 19%. The null hypothesis is a reasonable explanation of this data so we do not have strong evidence that the gambler can predict the outcome of the die. c. If the gambler had guessed correctly on forty of the rolls, would the p-value go up or down? Since this would be farther away from what is wexpected under the null, the p-value would get smaller.

Practice Question #2: Does the person paying the bill order more expensive meals, or less expensive meals, at a restaurant? The bills from 100 parties of two are examined and it turns out that the person paying the bill ordered a meal costing an average of $0.50 more than their companion with a standard deviation of $2.00 a. Clearly define the parameter of interest in this situation and then state the null and alternative hypotheses as statements about this parameter. μ = the average amount the bill-payer orders more than their companion for the whole population…null (Ho): μ = 0 alternative (Ha): μ ≠ 0

b. Find the test statistic and calculate the p-value. What do you conclude? x = 0.5 and z = (0.5-0)/(2.00/100 ) =2.5. From the tables the p-value ≈ 2(100-99.38) = 1.24%. It is unlikely that average differences of this size would occur for 100 pairs of customers. We have evidence that the person paying the bill and their companion do not order the same priced meals on average. c. In lecture we learned that statistical significance does not imply practical significance. Explain how this is illustrated by this example. Even though the bill-payer spent significantly more, the actual difference of 50 cents is very small and not likely to be of any practical importance. Practice Question #3: Which of these statements are true and which are false. Explain. A) If the p-value is 100% then the null hypothesis must be true. False – for a two-sided test this just says the data came out exactly as expected under the null hypothesis. But this might also happen under the alternate (though the chances would be lower). B) If the p-value is 1% then the alternative hypothesis has a 99% chance of being true. False – the p-value is calculated assuming the null hypothesis is true so it doesn’t tell you about the chances under the alternative. C) If the p-value is 50% this says that 50% of the time the test statistic would be this far from what is expected when the null hypothesis is true. True D) For the same data, a one-tailed significance test will give a lower p-value than a two-tailed significance test. True – since the p-value for the two-sided test includes the chance of getting both high and low values under the null. E) It is important to look at the results before deciding on whether to use a one-tailed or two tailed test. False – the null and alternative hypothesis are created before the data are collected. F) If you conduct 100 hypothesis tests you are likely to find some significant results even when the data are all just due to chance. True – for example about 5 of the tests will give p-values less than 5%.

Sampling Distributions and Confidence Interval Practice Problems. (note : you will need to use Table 21.1 and the table on page 435 of the text) Practice Question #1: A researcher wants to know the percentage of Columbus residents who would favor a two cent increase in the gasoline tax to fund road repairs.A random...
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