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- Migraine and Acupuncture: A migraine is a particularly painful type of headache, which patients sometimes wish to treat with acupuncture. To determine whether acupuncture relieves migraine pain, researchers conducted a randomized controlled study where 155 patients diagnosed with migraine headaches were randomly assigned to one of two groups: treatment or control. 79 patients in the treatment group received acupuncture that is specifically designed to treat migraines. 76 patients in the control group received placebo acupuncture (needle insertion at non-acupoint locations). 24 hours after patients received acupuncture, they were asked if they were pain free. Results are summarized in the contingency table below. Test to see if migraine pain relief is dependent on receiving acupuncture. Use a = 0.05. Treatment Control Total ∞º Pain Free: Yes Pain Free: No 20 6 26 a) What is the correct null hypothesis? O Ho: Migraine pain relief is dependent on receiving acupuncture. O Ho: P1 = P2 = P3…Bicycle frames can be made from carbon, steel, or othermaterials. Carbon frames are much more expensive thansteel, but they are also much lighter. Could this lighter frame have a significant impact on speed? In 2010 a doc-tor in England, who commutes daily to work 27 miles roundtrip on a bicycle, ran his own randomized experi-ment to investigate. Each day for several months Dr. Groves flipped a coin todetermine whether he would ride his steel frame bicycle orhis carbon frame bicycle to work and back. He rode a totalof 30 journeys on the steel frame bike and 26 journeys onhis carbon frame bicycle, recording his total commutingtime for each day. The summary statistics are shown below:a) Why would you prefer to see the actual data beforeproceeding with inference?b) Construct and interpret a 95% confidence interval forthe difference in average commuting times.c) Explain in this context what “95% confidence”means.d) Based on your confidence interval, do the doctor’sdata provide evidence…Omega-3 revisited Exercises 21 describes an experi-ment investigating a dietary approach to treating bipolar disorder. Researchers randomly assigned 30 subjects totwo treatment groups, one group taking a high dose ofomega-3 fats and the other a placebo.a) Why was it important to randomize in assigning thesubjects to the two groups?b) What would be the advantages and disadvantages ofusing 100 subjects instead of 30?
- List the three requirements for repeated trials of an experiment to constitute Bernoulli trials.According to the Rule of Three, when we have a sample size n with x=0 successes, we have 95% confidence that the true population proportion has an upper bound of 3/n. If 40 couples use a method of gender selection and each couple has a baby girl, what is the 95% upper bound for p, the proportion of all babies who are boys?Due to Covid-19 pandemic, the demand for disposable medical face mask has increased drastically. A particular brand of medical face mask making machine is used to produce disposable medical face masks in mass production. This machine is capable of producing 80 to 100 industry standard surgical style face masks every minute. This particular machine has a 2% defective rate. The production of medical face masks on this machine is considered a random process where each produced mask is independent of the others. Using this information calculate the following probabilities: a) What is the probability that the 10th medical face mask produced is the first with a defeci. b) What is the probability that the face mask making machine produces no defective mask in a batch of 100? ( ) c) On average, how many face masks would you expect to be produced before the first defective mask? What is the standard deviation? (. d) Another brand of this machine that also produces surgical style face masks has…
- A manufacturer of widgets wants to test a new widget-producing machine to determine if it can make an average of 25 widgets per second before deciding to invest in the machine. The standard to reject the new machine is if it makes an average of less than 25 widgets per second. Here are data from a small random sample:25.6, 26.2, 22.5, 20.5, 26.4, 27.4, 23.6, 26.9, 25.7, 24.9Assuming the population follows a normal distribution, is there evidence that the new machine should be rejected at the 0.01 significance level? State the hypotheses, list and check the conditions, calculate the test statistic, find the p-value, and make a conclusion in a complete sentence related to the scenario.In a completely randomized design experiment, 10 experimental units were randomly chosen for each of three treatment groups and a quantity was measured for each unit within each group. In the first steps of testing whether the means of the three groups are the same, the sum of squares for treatments was calculated to be 3,110 and the sum of squares for error was calculated to be 27,000. Complete the ANOVA table. Source df SS MS F Treatments Error TotalIn randomized, double-blind olinical trials of a new vaccine, rats were randomly divided into two groups. Subjects in group x received the new vaccine while subjects in group y received a control vaccine. After the first dose, 118 of 718 subjects in the experimental group (group x) experienced vamiting as a side effect. After the first dose, 74 of 619 of the subjects in the control group (group y) experienced vomiting as a side effect. Construct a 95% confidence interval for the difference between the two population proportions, P, - P, The 95% confidence interval for P,-P, is ( (Round to three decimal places as needed.)
- Late Night Eating It is well-known that lack of sleep impairs concentration and alertness, and this might be due partly to late night food consumption. A study74 took 44 people aged 21 to 50 and gave them unlimited access to food and drink during the day, but allowed them only 4 hours of sleep per night for three consecutive nights. On the fourth night, all participants again had to stay up until 4 am, but this time participants were randomized into two groups; one group was only given access to water from 10 pm until their bedtime at 4 am while the other group still had unlimited access to food and drink for all hours. The group forced to fast from 10 pm on performed significantly better on tests of reaction time and had fewer attention lapses than the group with access to late night food. What are the explanatory and response variables? Is this an observational study or a randomized experiment? Can we conclude that eating late at night worsens some of the typical effects of sleep…Female undergraduates in randomized groups of 15 took part in a self-esteem study. The study measured an index of self-esteem from the point of view of competence, social acceptance, and physical attractiveness. Let x1, x2, and x3 be random variables representing the measure of self-esteem through x1 (competence), x2 (social acceptance), and x3 (attractiveness). Higher index values mean a more positive influence on self- esteem. Variable Sample Mean Standard Deviation Population Size Мean 15 19.48 2.92 X1 X2 X3 (a) Find a 90% confidence interval for u1 - 42. (Round your answers to two decimal places.) lower limit 15 19.90 3.98 H2 15 17.42 3.60 upper limit (b) Find a 90% confidence interval for u1 - 43. (Round your answers to two decimal places.) lower limit upper limit (c) Find a 90% confidence interval for u2 - 13. (Round your answers to two decimal places.) lower limit upper limit
