Almost all medical schools in the United States require students to take the Medical College Admission Test (MCAT). To estimate the mean score μ of those who took the MCAT on your campus, you will obtain the scores of an SRS of students. The scores follow a Normal distribution, and from published information you know that the standard deviation is 10.4 . Suppose that, unknown to you, the mean score of those taking the MCAT on your campus is 500 .

In answering the questions, use z‑scores rounded to two decimal places.

(a) If you choose one student at random, what is the probability that the student's score is between 495 and 505 ? Use  Table A, or software to calculate your answer.

(Enter your answer rounded to four decimal places.)

probability:

(b) You sample 25 students. What is the standard deviation of the sampling distribution of their average score ¯x ? (Enter your answer rounded to two decimal places.)

standard deviation:

(c) What is the probability that the mean score of your sample is between 495 and 505 ? (Enter your answer rounded to four decimal places.)

probability:

In 1974, Loftus and Palmer conducted a classic study demonstrating how the language used to ask a question can influence eyewitness memory. In the study, college students watched a film of an automobile accident and then were asked questions about what they saw. One group was asked, “About how fast were the cars going when they smashed into each other?” Another group was asked the same question except the verb was changed to “hit” instead of “smashed into.” The “smashed into” group reported significantly higher estimates of speed than the “hit” group. You, as a researcher wonder if Loftus and Palmer’s study is reliable, and repeats this study with a sample of FIU students and obtains the following data.

Your job is to determine if smashed into group reports higher speed than hit group. As you work on this problem, make sure to provide information for each of the eight steps we cover in Chapter 11 (Salkind) as well as the APA write-up you would see in a results section.

1. State the null and alternative hypotheses
2. Tell me your level of risk
3. Determine the best statistical test to use
4. Determine the value needed to reject the null hypothesis. Remember to calculate the correct degrees of freedom before finding the critical t-value! Note whether it is best to use the one-tailed or two-tailed test.
5. Compare the obtained and critical value
6. Decide whether you will retain the null hypothesis or …
7. Decide whether you will reject the null hypothesis
8. Finally, write up your results as you would see it in a results section of an empirical research paper. Make sure to include the means and SDs for smashed into and hit group (in miles). I do NOT need to see the effect size (Cohen’s D)
9. Was your obtained t-value positive or negative? Would it matter either way? With your discussion group, tell my why a positive or negative value is not important when it comes to your obtained value
10. What is more appropriate to use for your data set: the one-tailed t-Test or the two-tailed t-Test. Why? Would your APA write-up differ depending on which you used?
11. Why would it be easier to find significance using a p value of .05 than a p value of .01?
12. Finally (and this is the tough one), how would your results have differed with regard to steps 4 through 9 if you had used n rather than n – 1?

What is the difference between a random sample and a simple random​ sample?

A service station has both self-service and full-service islands. On each island, there is a single regular unleaded pump with two hoses. Let X denote the number of hoses being used on the self-service island at a particular time, and let Y denote the number of hoses on the full-service island in use at that time. The joint pmf of X and Y appears in the accompanying tabulation.

(a) Given that X = 1, determine the conditional pmf of Y—i.e., p_Y|X(0|1), p_Y|X(1|1), p_Y|X(2|1). (Round your answers to four decimal places.)

(b) Given that two hoses are in use at the self-service island, what is the conditional pmf of the number of hoses in use on the full-service island? (Round your answers to four decimal places.)

(c) Use the result of part (b) to calculate the conditional probability P(Y ≤ 1 | X = 2). (Round your answer to four decimal places.)
P(Y ≤ 1 | X = 2) =  

(d) Given that two hoses are in use at the full-service island, what is the conditional pmf of the number in use at the self-service island? (Round your answers to four decimal places.)

Chi-Square test

Question 1   Data have been collected over a period of several months on the means of transportation used by most persons commuting to work into the city office of a large business firm. The results have been listed below:

Conduct a hypothesis test to determine if there is a significant difference among method of getting to work for this sample. Use a = 0.05.

Question 2 Influenza A is a virus which causes illness in humans and many other animal species. Once the virusfinds a host, it becomes a major concern and is fatal in some cases. A microbiologist wants to know if a person’s ageaffects the severity of conditions for people who contract Influenza A. Consider the table below and answer thequestions that follow.

a)  State the null and alternative hypotheses for this scenario.

1. Using the table above; calculate the expected values (two decimal places) for each outcome (write your answers in the table).

c)  Calculate the chi-square test statistic and degrees of freedom for this scenario.

d)  State your conclusion for this scenario (use a significance level of 0.01).

One of the tree planters on your team is a law student. You've been talking to her (from a safe distance) about how the hypothesis testing process works in psychology and, in particular, about the idea of setting an alpha level to .05 (1 in 20). She is appalled and responds: ``If I made an error one time in twenty there would be so many innocent people in jail it would be a major tragedy. You psychologists need to clean up your act and set your alpha level to something like one in 100,000 or one in a million''. Explain to your law student teammate why what she's suggesting would be a bad idea.

Part 1:

Using Excel’s Randbetween(0,9) function, generate 200 samples of five random numbers between 0 and 9, calculate the mean of each sample. Show me the list of the 200 means. Typically, they should look like: 4.8, 3.6, 4.4, 6.0, etc.

Part 2:

Using Excel, calculate the overall mean of the 200 sample means (the average of the averages). This should be around 4.5.

Part 3:   

Using Excel, calculate the standard error of the mean (SEM)  (i.e. the standard deviation of the 200 sample means). We established in the previous simulation that the population average is 4.5 and the standarddeviation of the population is 2.87.

Since the SEM=  ^σ= σ/√n. The SEM therefore is 1.28. Thus, the standard deviation of the 200 sample means should be approximately 1.28.

Part 4:

Using Excel, make the histogram of the 200 sample means (sampling distribution of the mean) (use interval size 1, i.e., 0-1, 1-2, 2-3, …8-9). According to the Central Limit Theorem a bell shaped curve should appear. Show me this graph.

Part 5:

Discuss the intuitive logic of the Central Limit Theorem. Discuss the implications of part 4 in this context.  (My videos might help here.)

Part 6:

Use 2 methods to find P (>6.3), (with n=5 as in Parts 1-4): First the z-method of chapter 7 and then by simply counting how many of your 200   were above 6.3.

Part 7:

Discuss the standard error of the mean.

1. Find the critical value(s) for a hypothesis test using a sample of size n = 12,
   a significance level α = 0.01 and null hypothesis H₀: μ ≤ 40.
   
   Assume that the population SD is unknown.

   		2.718
   		2.68
   		3.11
   		2.33

Assume workers transition through the labor force independently with the
transitions following a homogeneous Markov chain with three states:
• Employed full-time
• Employed part-time
• Unemployed
The transition matrix is:


0.90 0.07 0.03
0.05 0.80 0.15
0.15 0.15 0.70

.
• Worker Y is currently employed full-time
• Worker Z is currently employed part-time
Find the probability that either Y or Z, but not both will be unemployed after two transitions.

Suppose a sample of 455 people is drawn. Of these people, 254 passed out at G forces greater than 6. Using the data, estimate the proportion of people who pass out at more than 6 Gs. Enter your answer as a fraction or a decimal number rounded to three decimal places.

According to a Gallup poll about gun ownership, in the year 2016, 270 out of 600 (45%) U.S. households answered “yes” to the question: “Do you have a gun in your home?”.

a. List the requirements for constructing a confidence interval for a proportion and show how the requirements are met for this problem.

b. Construct a 95% confidence interval for the proportion of households who own a gun in the year 2016.

c. Interpret your confidence interval in part “c”. (I am ____% confident that………).

d. Sample Size: A politician wants to know if the proportion of U.S. households who own a gun is on the rise. What size sample should be obtained if the politician wants an estimate within 3 percentage points of the true proportion with 95% confidence if he uses the 2016 estimate of 37.7% (use formula pg. 401)?

2. A group of 24 physical doctors from three countries consists of 7 Americans, 8 British and 9 Chinese. They (wearing masks) are seated at random around a (large enough) table to discuss the means of treatment of COVID-19 patients. Compute the following probabilities:

(a) that all the Americans sit together; (b) that all the Americans and the British sit together; (c) that all the Americans, all the British and all the Chinese sit together.

(d) Randomly choose 6 doctors from the group of doctors in the above question to evaluate the efficiency of one type of treatment of COVID-19. It is unfair if all of the 6 doctors are from the same country. What is the probability that the selection is fair?

Please show work, write legibly!

1. The researcher from the Annenberg School of Communications is interested in studying the factors that influence how much time people spend talking on their smartphones. She believes that gender might be one factor that influences phone conversation time. She specifically hypothesizes that women and men spend different amounts of time talking on their phones. The researcher conducts a new study and obtains data from a random sample of adults from two groups identified as women and men. She finds that the average daily phone talking time among 15 women in her sample is 42 minutes (with a standard deviation of 6). The average daily minutes spent talking on the phone among 17 men in her sample is 38 (with a standard deviation of 5). She selects a 95% confidence level as appropriate to test the null hypothesis.

a) Please identify the independent variable for the researcher's hypothesis in the text box below.

b) What is the unit of analysis?

c) What is the alpha?

d) State the research and null hypothesis in symbols. Make sure to be as complete as possible (Using H1: and H0:).

The following data represent the age in weeks at which babies first crawl based on a survey of 12 mothers conducted by Essential Baby. 52 30 44 35 39 26 47 37 56 26 39 28

a. Draw a normal probability plot (use Statcrunch) and boxplot to determine if it is reasonable to conclude the data come from a population that is normally distributed. Copy/paste the graphs into your solution along with an explanation of why or why not.

b. Construct a 90% confidence interval for the mean age at which a baby first crawls.

c. Interpret your confidence interval in part “b”. (I am ____% confident that ______).

d. Sample Size: How large a sample size is needed to estimate the mean age in weeks at which a baby first crawls within 1.5 weeks with 95% confidence? (use formula on pg. 413)