Subject: Probability and Statistics

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The owner of a factory that sells particular bottled water claims that the average capacity of their product is 500 mL. To test the claim, a consumer group gets a sample of 100 such plastic bottles, calculate the capacity of each bottle, and then finds the mean capacity to be 497 mL. The standard deviation is 5 mL. Is the claim true? Note: Use the seven-step method.GENERALIZATIONS EXAMPLE 3 A researcher used a developed problem solving test to randomly select 50 Grade 11 students. In this sample, x = 90 and s=10. The mean u and the standard deviation of the population used in the standardized of the test were 85 and 15, respectively. Use the 95%% confidence level to answer the following questions: Does the sample mean differ significantly from the population mean? Steps Answer 1. Describe the population The parameter Interest is the mean parameter of interest, u of the population where the sample comes from: 2. Formulate the hypotheses: the Hail = 65 null and the alternative hypothesis. That Is, state a null Hjiu = 65 hypothesis, Ho In such a way that Type | error can be calculated. 3. Check the assumptions. Since n=50, by the Central Limit Is the sample size large enough Theorem, the distributions is for the Central Limit Theorem normally distributed. (CLT) to apply? .Do small samples come from (This assumptions need not to be normally distributed populations? addressed) "Are the sample selected randomly? Yes! 4. Choose the significance level a =1-0.95 =0.05 for a. Two-tailed .Is the test two-tailed or one- Z critical values: + 1.96 tailed? .Get the critical value from the test statistic table. .Establish the critical regions.5. Select the appropriate Test statistic: z and o -15 test statistic. "Compute the test statistic. 90 - 85 5 = 2.36 15 2.121320 50 6. State the decision rule. Reject Ho If the computed test statistic S Negative critical value or if the computed test statistic _ positive critical value. Otherwise, do not reject (or accept) Ho 7. Compare the test Decision making:" statistic and critical value. 2.36 5 1.96 .Based on the decision rule, decide whether to The null hypothesis is rejected. reject or to accept Ho . "Interpretations .Interpret the result. .There Is enough evidence to reject the null hypothesis Take a course of action .There Is a significant difference between (optional) the sample mean and the population mean