Flaal Project Fall 2023 Math 153 Statistics Bettling Company Case Study Imagine you are a manager at a major bottling company. Customers have begun to complain that the bottles of the brand of soda produced in your company contain less than the advertised sixteen (16) ounces of product. Your boss wants to solve the problem at hand and has asked you to investigate. You have your employees pull thirty (30) bottles off the line at random from all the shifts at the bottling plant. You ask your employees to measure the amount of soda there is in each bottle. Note: Use the data set provided in the following table to complete this assignment Bottle Ounces Bottle Ounces Number Number Bottle Number Ounces 1 14.5 11 15 21 14.1 2 14.6 12 15.1 22 14.2 3 14.7 13 15 23 14 4 14.8 14 14.4 24 14.9 5 14.9 15 15.8 25 14,7 6 15.3 16 14 26 14.5 7 14.9 17 16 27 14.6 8 15.5 18 16.1 28 14.8 9 14.8 19 15.8 29 14.8 10 15.2 20 14.5 30 14.6 Write a two to three (2-3) page report in which you: 1. Calculate the mean, median, and standard deviation for ounces in the bottles. 2. Construct a 95% Confidence Interval for the ounces in the bottles. 3. Conduct a hypothesis test to verify if the claim that a bottle contains less than sixteen (16) ounces is supported. Clearly state the logic of your test, the calculations, and the conclusion of your test. 4. Provide the following discussion based on the conclusion of your test: a. If you conclude that there are less than sixteen (16) ounces in a bottle of soda, speculate on three (3) possible causes. Next, suggest the strategies to avoid the deficit in the future
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The results were analyzed in R via a 2-sample t-test with a 2-sided alternative hypothesis, and some of the output is summarized below (assume ̑ = 0.05 where necessary). mean sd n Alumania 3.67 3.34 35 Beautinium 4.01 3.74 35 Two Sample t-test data: impurity by supplier t = -0.413, df = 68, p-value = 0.6895 alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: -2.03 1.35 Levene's Test for Homogeneity of Variance (center = "median") Df F value Pr(>F) group 1 0.1977 0.658 68 Let ̑1 and ̑2 denote the mean amount of impurities from the suppliers “Alumania‐ and “Beautinium" respectively, and ̑1^2 and ̑2^2 be the respective variances. If you were asked to use this analysis to answer the question of which supplier of aluminium to use, what is your conclusion? a. Beautinium has a higher sample mean of impurities, so we should use Beautinium. b. p = 0.6895> 0.05 so we cannot reject H0, ie there is no significant evidence that there is any difference in the amount of impurities produced by the two suppliers. c. Aluminia has a lower sample mean of impurities, so we should use Aluminia. d. p = 0.658 > 0.05 so we cannot reject H0, ie there is no significant evidence that there is any difference in the amount of impurities produced by the two suppliers.
Kari H.
a) It is reported that scores on a national accounting certification exam are approximately normally distributed with a mean of 78. Professor Rose believes that students from her three sections of her preparatory course will score significantly higher than the national average on this test. After they write the exam, her students have scored an average of 84. After conducting the appropriate statistical test, Professor Rose finds that the p-value is 0.0022 Which of the following is the best interpretation of the p-value? A p-value of 0.0022 indicates that there is a very small chance that Professor Rose's class outperformed the other exam writers across the nation. A p-value of 0.0022 provides strong evidence that Professor Rose's class outperformed the other exam writers across the nation. A p-value of 0.0022 provides evidence that Professor Rose is an exceptional teacher who was able to prepare her students well for this exam. None of the above are reasonable interpretations of this result. b) Food inspectors inspect samples of Alberta beef to see if they are safe. This can be thought of as a hypothesis test where H0: the beef is safe H1: the beef is not safe. At one plant, the inspector concludes the beef is safe when it actually is not safe. Which of the following statements is correct? The inspector has made a Type I error. The inspector has made a Type II error. It is impossible to know which of the two types of errors has been made without knowing the level of significance. It is impossible to know which of the two types of errors has been made without knowing the p-value. c) Prairie Craft Brewery has an automated filling machine that is set to fill each bottle with 500 millilitres of the its new Redhead Rye Ale. They have just completed the testing on a sample of 180 bottles to to see if the machine is filling the bottles with an average of 500 millilitres of ale. The tester has obtained a mean of 510 ml and p-value of 0.05 for a right-tail test. Which definition of the p-value is the most accurate? It is the probability that the observed mean of 510 ml will occur again. It is the probability that the null hypothesis is true. It is the probability of observing a mean of 510 ml or more under the assumption that the null hypothesis is true. It is the value that any sample mean must reach in order to be considered significant under the null hypothesis. d) An entrepreneurial scientist has invented a supplement that she believes can enhance short-term memory. She conducts an experiment to test the effect of the supplement on 23 volunteers, 8 of whom are given a placebo and 15 of whom are given the supplement. In analyzing in her study, she obtains a p-value of 0.24. Which of the following is a reasonable interpretation of her results? She should reject the null hypothesis. She should fail to reject the null hypothesis. This proves that at least 24% of the time, the experimental treatment has no effect on memory. This proves that at most 24% of the time, the experimental treatment has no effect on memory. e) A human resources study claims that the average age of Canadians who retire is 66 years. You believe that the average age is less than that. You take a random sample of 15 Canadians who have retired, and find their average age to be 65.4 years, with a standard deviation of 0.80 years. Using the resulting p-value of 0.0017, you conclude that this is significantly lower than the age of 66 stated in the study. What would be the most appropriate interpretation of this result? An error must have been made. This difference is too small to be statistically significant. Without knowing the level of significance, it is impossible to know if the results are statistically significant. Although the result is statistically significant, the difference in age is of little practical importance. The statistically significant result indicates that the majority of people who retire are younger than 66.
Jon S.
A. In a clinical trial, 21 out of 842 patients taking a prescription drug daily complained of flu-like symptoms. Suppose that it is known that 2.2% of patients taking competing drugs complain of flu-like symptoms. Is there sufficient evidence to conclude that more than 2.2% of this drug's users experience flu-like symptoms as a side effect at the α=0.01 level of significance? Find t-stat, Z0, and p-value. B. Suppose a mutual fund qualifies as having moderate risk if the standard deviation of its monthly rate of return is less than 5%. A mutual fund rating agency randomly selects 21 months and determines the rate of return for a certain fund. The standard deviation of the rate of return is computed to be 3.67%. Is there sufficient evidence to conclude that the fund has moderate risk at the α=0.10 level of significance? A normal probability plot indicates that the monthly rates of return are normally distributed. Find p-value. C. The piston diameter of a certain hand pump is 0.6 inches. The manager determines that the diameters are normally distributed, with a mean of 0.6 inches and a standard deviation of 0.003 inches. After recalibrating the production machine, the manager randomly selects 29 pistons and determines that the standard deviation is 0.0022 inches. Is there significant evidence for the manager to conclude that the standard deviation has decreased at the α=0.10 level of significance? Find p-value. D. Suppose a professional golfing association requires that the standard deviation of the diameter of a golf ball be less than 0.004 inches. Determine whether these golf balls conform to this requirement at the α=0.05 level of significance. Assume that the population is normally distributed. Find sample standard deviation, use s to find t statistic, and find p-value. E. A psychologist obtains a random sample of 20 mothers in the first trimester of their pregnancy. The mothers are asked to play Mozart in the house at least 30 minutes each day until they give birth. After 5 years, the child is administered an IQ test. It is known that IQs are normally distributed with a mean of 100. If the IQs of the 20 children in the study result in a sample mean of 104.4 and sample standard deviation of 16, is there evidence that the children have higher IQs? Use the α=0.05 level of significance. Complete parts (a) through (d). Find p-value. F. In 1945, an organization asked 1536 randomly sampled American citizens, "Do you think we can develop a way to protect ourselves from atomic bombs in case others tried to use them against us?" with 821 responding yes. Did a majority of the citizens feel the country could develop a way to protect itself from atomic bombs in 1945? Use the α=0.05 level of significance. Find p-value. G. A manufacturer of alloy steel beams requires that the standard deviation of yield strength not exceed 7000 psi. The quality-control manager selected a sample of 20 beams and measured their yield strength. The standard deviation of the sample was 8100 psi. Assume that the yield strengths are normally distributed. Does the evidence suggest that the standard deviation of yield strength exceeds 7000 psi at the α=0.01 level of significance? Complete parts (a) through (d) below. Find p-value.
Sri K.
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