Table 1 shows data on whether or not a treatment relieved pain for patients. Treatment Placebo Total Relieved pain 41 21 62 Did not relieve 39 54 93 pain Total 80 75 155 Table 1 Is the treatment significantly better than the placebo? Test whether the treatment is significantly better than a placebo at relieving pain. The patients were randomly allocated to the two groups and the experiment was double-blind.
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Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
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- Elizon Pharmaceuticals, Inc., announced the results of its first human trial of NPI 32101, a topical form of its skin ointment. A total of 225 patients diagnosed with skin irritations were randomly divided into three groups as part of a double-blind, placebo-controlled study to test the effectiveness of the new topical cream. The first group received a 0.5% cream, the second group received a 1.0% cream, and the third group received a placebo. Groups were treated twice daily for a 6-week period. (a)What type of experimental design is this? (b)What is the control group for this study? (c)What is the factor that is set to predetermined levels? What is the treatment? (d)What does it mean for this study to be double-blind? (e)Identify the experimental units. (f)Draw a diagram to illustrate the design.A study was performed on 200 elementary school students to investigate whether regular Vitamin A supplementation was effective in preventing colds during the month of March. 100 were randomized to receive daily Vitamin A supplements during the month of March, and 100 students were randomized to a placebo group (and did not receive Vitamin A) during the same month. The number of students getting at least one cold in March was computed in the two groups, and the results are given in the following 2 X 2 table. Using a 5% level of significance determine whether there is an association between Vitamin A supplementation and prevention of Common Cold ColdNo Cold Vitamin A1585100 Placebo2575100 40160200A researcher is interested in estimating the extent to which posting caloric labels on menus changes the number of calories consumed by patrons. To test the impact of posting caloric labels on menu items, the analyst created two menus for the same restaurant: One menu with caloric labels and one menu without. As soon as restaurant guests entered the restaurant, they were randomly assigned one of the two types of menus. For the sake of clarity, guests assigned the menu with caloric labels are referred to as “Group 1” or the “treatment group”. Likewise, guests assigned the menu without caloric labels are referred to as “Group 2” or the control group. The analyst then collected data detailing how many calories each guest consumed as well as recording the age of each guest. The variable Cal1 denotes the number of calories consumed by guests who were assigned to the treatment group 1, while the variable Cal2 denotes the number of calories consumed by guests who were assigned to the control…
- A consumer advocacy group wanted to study whether different airline carriers differed in terms of their delayed flights. In particular, the researchers were interested in the relationship between p1, the proportion of Alpha Airlines flights that were delayed at least 15 minutes, and p2, the proportion of Beta Airlines flights that were delayed at least 15 minutes. A random sample of 1,000 Alpha flights and a separate random sample of 1,000 Beta flights found that 67 of the Alpha fights and 160 of the Beta flights were delayed at least 15 minutes. The conditions for inference were checked and verified. Does this set of samples provide strong evidence that Alpha Airlines has a smaller proportion of flights that are delayed at least 15 minutes than Beta Airlines, at the α = 0.05 significance level? Find the z-table here. A. The test statistic is z = –6.56 and the P-value ≈ 0. Since the P-value ≈ 0 < 0.05, there is not sufficient evidence that Alpha Airlines has fewer delayed…A poll asked 946 adults "If purchasing a used car made certain upgrades or features more affordable, what would be your preferred luxury upgrade? The results indicated that 8% of the males and 17% of the females answered window tinting. The poll description did not state the sample sizes of males and females. Suppose that both sample sizes were 473 and that 40 of 473 males and 80 of 473 females reported window tinting as their preferred luxury upgrade of choice. b. Find the p-value in (a) and interpret its meaning. The p-value is _______ (Round to three decimal places as needed.) Interpret the p-value. The p-value is the probability that in a random sample of 946 adults, the difference between the proportions of males and females who said they prefer window tinting as a luxury upgrade will be _________the difference in this sample, assuming that the proportion of males who said they prefer window tinting as a luxury upgrade is ________the proportion for females…A survey of 900 adults from a certain region asked, "If purchasing a used car made certain upgrades or features more affordable, what would be your preferred luxury upgrade?" The results indicated that 57% of the females and 46% of the males answered window tinting. The sample sizes of males and females were not provided. Suppose that of 300 females, 171 reported window tinting as their preferred luxury upgrade of choice, while of 600males, 276 reported window tinting as their preferred luxury upgrade of choice. a. Determine the zstat, and the p-value. b.Determine the critical values and a conclusion c. Construct a 95% confidence interval estimate of the difference between the two population proportions.
- According to a certain government agency for a large country, the proportion of fatal traffic accidents in the country in which the driver had a positive blood alcohol concentration (BAC) is 0.33.Suppose a random sample of 109 traffic fatalities in a certain region results in 47that involved a positive BAC. Does the sample evidence suggest that the region has a higher proportion of traffic fatalities involving a positive BAC than the country at the α=0.05 level of significance?A study is made of residents in Phoenix and its suburbs concerning the proportion of residents who subscribe to Sporting News. A random sample of 90 urban residents showed that 12 subscribed, and a random sample of 97 suburban residents showed that 17 subscribed. Does this indicate that a higher proportion of suburban residents subscribe to Sporting News? Use a 1% level of significanceIn a randomized double-blind clinical trials of a new vaccine children were randomly divided into two groups subjects in group one received the new vaccine while subjects in group 2 received a control vaccine after the second dose 112 of 690 subjects in the experimental group 1 experienced drowsiness as a side effect after the second dose 69 of 571 of the subjects in the control group 2 experienced drowsiness as a side effect does the evidence suggest that a higher proportion of subjects in group 1 experience drowsiness as a side effect than subjects in group 2 at 0.05 level of significance? Determine the null and alternative hypothesis? find the test statistic for this hypothesis test? determine the P value for this hypothesis test? do we reject or not reject the null hypothesis and why? State your conclusion?
- An article from the American Journal of Public Health reports the results from a randomized study designed to evaluate the efficacy of an intervention targeted to Hispanic/Latino men who identify as gay, bi-sexual or other men who have sex with men (MSM). A representative sample of 254 such men was randomized to be in either the intervention group (n=152, with 141 ultimately participating in the study) or the control group (n=152, with 147 ulitmately participating in the study). The primary outcome under study getting tested for HIV within the six-months following group assignment (randomization) among those who had been sexually active in this same six-month follow-up period. At six months of follow-up, 141 subjects in the intervention group reported having had sex (with men and/or women) since randomization. Of these 141 men, 114 had been tested for HIV since being randomized. At six months of follow-up, 147 subjects in the control group reported having had sex (with men…A study was conducted to determine whether automobile repair charges are higher for female customers than for male customers. Twenty auto repair shops were randomly selected from the telephone book. Two cars of the same age, brand, and engine problem were used in the study. For each repair shop, the two cars were randomly assigned to a man and woman participant and then taken to the shop for an estimate of repair cost. The repair costs (in dollars) are given in data 4. Which procedure, t or Wilcoxon, is more appropriate in this situation? Why? Are repair costs generally higher for female customers than for male customers? data4 'RepairShop' 'Female' 'Male'1 871 7922 684 7653 795 5114 838 5205 1033 6186 917 4477 1047 5488 723 7209 1179…Twenty middle-aged men with glucose readings between 90 milligrams per deciliter and 120 milligrams per deciliter of blood were selected randomly from the population of similar male patients at a large local hospital. Ten of the 20 men were assigned randomly to group A and received a placebo. The other 10 men were assigned to group B and received a new glucose drug. After two months, posttreatment glucose readings were taken for all 20 men and were compared with pretreatment readings. The reduction in glucose level (Pretreatment reading − Posttreatment reading) for each man in the study is shown here. Group A (placebo) reduction (in milligrams per deciliter): 12, 8, 17, 7, 20, 2, 5, 9, 12, 6 Group B (glucose drug) reduction (in milligrams per deciliter): 29, 31, 13, 19, 21, 5, 24, 12, 8, 21 Part A: Do the data provide convincing evidence, at the α = 0.02 level, that the glucose drug is effective in producing a reduction in mean glucose level? Part B: Create and interpret a 98%…