A retailer discovers that 3 jars from his last shipment of Spiffy peanut butter contained between 15.85 and 15.92 oz of peanut butter, despite the labeling indicating that each jar should contain 16 oz. of peanut butter. He is wondering if Spiffy is cheating its customers by filling its jars with less product than advertised. He decides to measure the weight of 50 jars from the shipment and use hypothesis testing to verify this.

(a) What are the null and alternative hypotheses for this experiment?

(b) Describe, in words, a Type I error for this experiment.

(c) Describe, in words, a Type II error for this experiment.

(d) Given the answer to (a), should the null hypothesis be rejected when the sample mean falls below or over a certain threshold? Should this threshold be below or above the value 16.0 oz?

(e) What is the distribution of X¯, the sample mean?

(f) In his sample of 50 jars, the retailer finds an average weight of 15.84 oz and a sample standard deviation of 0.5 oz. He decides to use a significance level of 0.04. What is the conclusion from this hypothesis testing? Can you conclude that Spiffy is cheating its customers?

(g) What is the p-value? What is the meaning of this number?

(h) For what values of the sample mean would the null hypothesis be rejected?

(i) Calculate the probability of type II error if the true mean is 15.7 oz.

(j) Solve (f), (h) and (i) when the level of significance is 0.01. Is your new answer for (f) consistent with the p-value found in (g)? How is the probability of type II error affected when the probability of type I error changes?

Given below are pulse rates for a placebo group, a group of men treated with Xynamine in 10 mg doses, and a group of men treated with 20 mg doses of Xynamine. The project manager for the drug conducts research and finds that for adult males, pulse rates are normally distributed with a mean around 70 beats per minute and a standard deviation of approximately 11 beats per minute. His summary report states that the drug is effective, based on this evidence: The placebo group has a mean pulse rate of 68.9, which is close to the value of 70 beats per minute for adult males in general, but the group treated with 10 mg doses of Xynamine has a lower mean pulse rate of 66.2, and the group treated with 20 mg doses of Xynamine has the lowest mean pulse rate of 65.2. Analyze the Results 1. Analyze the data using the methods learned in course. 2. Based on the results, does it appear that there is sufficient evidence to support the claim that the drug lowers pulse rates? 3. Are there any serious problems with the design of the experiment? 4. Given that only males were involved in the experiment, do the results also apply to females? 5. The project manger compared the post-treatment pulse rates to the mean pulse rate for adult males. Is there a better way to measure the drug’s effectiveness in lowering pulse rates? 6. How would you characterize the overall validity of the experiment? 7. Based on the available results, should the drug be approved? 8. Write a brief report summarizing your findings.

Placebo group:

77,61,66,63,81,66,79,66,75,48,70

10 mg group:

67,48,79,67,57,66,85,75,77,57,45

20 mg group:

72,94,57,63,69,64,82,34,76,59,53

The purpose of this experiment is to prepare tetraamminecopper (II) sulfate monohydrate by making a reaction between ammonia and copper (II) sulfate pentahydrate in an aqueous solution. When the reaction is complete, the product is then isolated and percent yield is determined. We had to weigh CuSO₄ 5H₂O and then mix it with water then heat. Ammonia is then added once the solid dissolves and is heated to a gentle boil. The solution is then put into a buchner funnel and drained of all liquid with a water aspriator. The solid remaining is cleaned off with ammonia and ethanol solution and then just pure ethanol. The solid remaining is weighed. 1. What did you expect to happen and what actually happened? Why did it

2. What is your interpretation of the physical and chemical changes observed

during the experiment?

3. What type of chemical reaction was performed? Is there any physical evidence

that the reaction happened?

4. How did you know the reaction was over?

5. Considering the collected data, identify the type of problem that you’re working

with (simple stoichiometry, limiting reactant and/or percent yield) and describe

the steps needed to solve it. Integrate your calculations to show how identifying

the problem guides your solution.

6. Were you able to identify the limiting reagent?

7. What is the meaning of the % yield obtained? Could it be better?

8. What is your interpretation of the chemical equation (think about mole

9. Will the opposite reaction take place (water replacing ammonia ligands)?

10. Explain (Consider the properties of the chemicals involved and the nature of the

observed chemical reaction)

11. Could this reaction be performed in the solid phase? Why solutions are needed?

12. How could you improve the results?

13. Any other topics you consider important to this experiment.

People who eat lots of fruits and vegetables have lower rates of colon cancer than those who eat little of these foods. Fruits and vegetables are rich in "antioxidants" such as vitamins A, C, and E. Will taking antioxidants help prevent colon cancer? A medical experiment studied this question with 864 people who were at risk of colon cancer. The subjects were divided into four groups: daily beta-carotene, daily vitamins C and E, all three vitamins every day, or daily placebo. After four years, the researchers were surprised to find no significant difference in colon cancer among the groups.

Outline the design of the experiment. Use your judgment in choosing the group sizes. (Select all that apply.)

Randomly assign 2 treatments to 4 groups.

Randomly assign subjects to 4 treatments.

Observe the occurrence of colon cancer.

Randomly assign subjects to 2 treatments.

Observe eating habits.

Suggest some possible reasons (lurking variables) that could explain why people who eat lots of fruits and vegetables have lower rates of colon cancer. The experiment suggests that these variables, rather than the antioxidants, may be responsible for the observed benefits of fruits and vegetables. (Select all that apply.)

People who eat lots of fruits and vegetables may have healthier diets overall.
People who eat more fruits and vegetables may exercise more.
People who eat lots of fruits and vegetables may watch more TV.
Fruits and vegetables contain fiber and this could account for the benefits of those foods.
Fruits and vegetables do not contain any sugar and this could account for the benefits of those foods.

Experiment 1: Study the Relationship Between Volume and Temperature for a Sample of Air

Took a Erlenmeyer flask & placed it on the workbench; closed flask.

Added 1.50 atm of air to the flask.

Attached a gas syringe & thermometer to the Erlenmeyer flask; remember the Erlenmeyer flask volume is 150 mL.

The volume of air in the syringe = 75.0 mL

The total air volume = 225.0 mL

The temperature = 21.5 C

Ran the constant temperature bath to 0 °C; moved the Erlenmeyer flask into the constant temperature bath. Waited until the temperature stabilized at 0.0 °C.

The volume of air in the syringe = 58.6 mL

The total air volume = 208.6 mL

The temperature = 0.0 °C.

Ran the bath to 40 °C.

Waited until the temperature of the air stabilized at 40.0 °C.

The volume of air in the syringe = 89.1 mL

The total air volume = 239.1 mL

The temperature = 40 °C.

Ran the bath to 60 °C

The volume of air in the syringe = 104.4 mL

The total air volume = 254.4 mL

The temperature = 60 °C

Ran the bath to 80 °C

The volume of air in the syringe = 119.7 mL

The total air volume = 269.7 mL

The temperature = 80 °C

Ran the bath to 100 °C

The volume of air in the syringe = 135.0 mL

The total air volume = 285.0

What parameters were held constant in experiment 1?

A. pressure and number of moles

B. volume and pressure

C. temperature and volume

D. volume and number of moles

Given the data you collected in experiment 1, what is the relationship between volume & temperature?

A. The volume increases as the temperature increases.

B. The volume increases as the temperature decreases.

C. The volume changes randomly with temperature.

D. The volume remains constant as the temperature increases.

A retailer discovers that 3 jars from his last shipment of Spiffy peanut butter contained between 15.85 and 15.92 oz of peanut butter, despite the labeling indicating that each jar should contain 16 oz. of peanut butter. He is wondering if Spiffy is cheating its customers by filling its jars with less product than advertised. He decides to measure the weight of 50 jars from the shipment and use hypothesis testing to verify this.

(a) What are the null and alternative hypotheses for this experiment?

(b) Describe, in words, a Type I error for this experiment.

(c) Describe, in words, a Type II error for this experiment.

(d) Given the answer to (a), should the null hypothesis be rejected when the sample mean falls below or over a certain threshold? Should this threshold be below or above the value 16.0 oz?

(e) What is the distribution of X ̄, the sample mean?

(f) In his sample of 50 jars, the retailer finds an average weight of 15.84 oz and a sample standard deviation of 0.5 oz. He decides to use a significance level of 0.04. What is the conclusion from this hypothesis testing? Can you conclude that Spiffy is cheating its customers?

(g) What is the p-value? What is the meaning of this number?

(h) For what values of the sample mean would the null hypothesis be rejected?

(i) Calculate the probability of type II error if the true mean is 15.7 oz.

(j) Solve (f), (h) and (i) when the level of significance is 0.01. Is your new answer for (f) consistent with the p-value found in (g)? How is the probability of type II error affected when the probability of type I error changes?

There is an urban legend that mothers have increased sensitivity to and awareness of noises, in particular to that of children. A social psychologist finds support for the legend in the literature and now wants to confirm it. The psychologist designs a study where he recruits women that think they are going to participate in a sleep experiment where they will evaluate the comfort of different mattresses overnight. While the women slept, the psychologist introduces noises to test the minimum volume needed for the women to be awakened by the noise. The wake-up decibel data are below. What can be concluded with α = 0.10?

b)
Condition 1: (choose one)
1) non-mother 2) sleep experiment 3) decibels 4) women 5) mother
Condition 2: (choose one)
1) non-mother 2) sleep experiment 3) decibels 4) women 5) mother

c) Input the appropriate value(s) to make a decision about H₀.
(Hint: Make sure to write down the null and alternative hypotheses to help solve the problem.)
p-value =_________________________ ; Decision: Reject H0 or Fail to reject H0

d) Using the SPSS results, compute the corresponding effect size(s) and indicate magnitude(s).
If not appropriate, input and/or select "na" below.
d =___________ ; ( choose one) 1.) trivial effect 2) small effect 3) medium effect 4) large effect
r² = ___________ ; ( Choose one) 1.) trivial effect 2) small effect 3) medium effect 4) large effect

Q3. (This question is based in R)
Now use the simulation ("X = rnorm(1000, mean = 10, sd = 2)", "Y = rnorm(1000, mean = 5, sd = 3)") to estimate the distribution of X+Y and create confidence intervals.

A) Form a set of Xs and Ys by repeating the individual experiment for B = 2000 times, where each experiment has n = 1000 samples. You may want to write a for loop and create two matrices "sample_X" and "sample_Y" to save those values. 

B) Calculate the mean of X+Y for each experiment and save it to a vector which has a length of B, and plot a histogram of these means. 

C) Now as we have a simulated sampling distribution of X+Y, calculate the 95% confidence interval for mean of X+Y (this can be done empirically). 

D) In the above example, we have fixed the sample size n and number of experiments B. Next, we want to change B and n, and see how the confidence interval will change. Please write a function to create confidence intervals for any B and n. 

E) Suppose the sample size n varies (100, 200, 300, .... , 1000) (fix B=2000) and the number of experiments B varies (1000, 2000, ... , 10000) (fix n=500). Plot your confidence intervals to compare the effect of changing the sample size n and changing the number of simulation replications B. What do you conclude? (Hint: Check function errbar() in Hmisc package for plot - library(Hmisc))

fix n, B varies

fix B, n varies

Question 1: Determine whether the given description to an observational study or an experiment.

In a study of 424 children with particular disease, the subjects were monitored with an EEG while asleep.

Does the given description correspond to an observational study or an experiment?

A. The given description corresponds to an experiment.

B. The given description corresponds to an observational study.

C. The given description does not provide enough information to answer this question.

Question 2: Identify the type of sampling used (random, systematic, convenience, stratified, or cluster sampling) in the situation described below.

A man is selected by a marketing company to participate in a paid focus group. The company says that the man was selected because he was randomly chosen from all men in his tax bracket.

Which type of sampling did the marketing company use?

A. Systematic sampling, B. Random Sampling, C. Convenience Sampling, D. Cluster sampling, E. Stratified sampling

Question 3: Identify which type of sampling is used: random, systematic, convenience, stratified, or cluster.

A radio station asks its listeners to call in their opinion regarding the format of the morning show.

Which type of sampling is used?

A. Cluster, B. Stratified, C. Random, D. Systematic, E. Convenience

Question 4: Identify the type of sampling used (random, systematic, convenience, stratified, or cluster sampling) in the situation described below.

A woman experienced a tax credit. The tax department claimed that the woman was audited because everyone in four randomly selected districts was being audited.

Which type of sampling did the tax department use?

A. Cluster sampling

B. Systematic Sampling

C. Convenience Sampling

D. Random Sampling

E. Stratified Sampling

Runiowa is a fashion shoe company that tries to manufacture much more durable heels in 2020. The management team of Runiowa suggests two rubber materials A and B and the research team of Runiowa is asked to design an experiment to gauge whether the rubber A is more durable than the rubber B. 300 people in the US aged between 18 and 65 were randomly chosen. The rubber A is allocated at random to the right shoe or the left shoe of each individual. Then, the rubber B has been assigned to the other. For example, if Mr. Nathaniel is one of 300 people randomly chosen, then the right heel of Mr. Nathaniel is randomly assigned to be made with the rubber A and then his left heel is to be made with the rubber B. The research team measures the amounts of heel wear both the rubber A (wA) and the rubber B (wB) in each individual and records the difference wA − wB of 300 individuals. Even though the individuals are heterogeneous with different heights and weights, those individual heterogeneities will not obscure the comparison of treatment groups by focusing on the paired differences of each individual. Also as long as the heel materials are randomly assigned for each individual, there has been no restrictions on shoe styles. Note that the age of subjects is ranging from 18 to 65. In this way, researchers compare treatments within blocks controlling heterogeneity of individuals. The research team also repeats this experiment design with 300 people in the US aged between 18 and 65 chosen at random.

Question:

Is there a conjecture?

What is the response variable?

What is the explanatory variable?

What levels of the factor(s) were used in the expereiment?

What are the treatments for this experiment?

What are the experimental units?

What is the control?

Hoe much replication was used?

How was randomization used?