Two-way ANOVA (8 pts)

Research Scenario: A developmental psychologist is looking into the effects of play group size on empathy and whether this differs by age. She has two age groups – four year olds and eight year olds. Each of those age groups is split into three separate groups for an entire semester during structured free time: play alone (n = 8 per age group), dyads (n = 8 per age group), or groups of 4 (n = 8 per age group). At the end of the semester, empathy is measured using the Young Children’s Empathy Measure during the parent-teacher conference (Poresky, 1990), which uses an accuracy scale of 0 – 4, with higher scores indicating more cognitive and affective empathy.

The scores for each group are shown in the table below. Remember to name and define your variables (your two factors and your one dependent variable) under the “Variable View,” then return to the “Data View” to enter and analyze the data. Remember, each factor (i.e. independent variable) is represented within a single column in SPSS using numbers to identify the categories (e.g., 0 = four year olds / 1 = eight year olds). The dependent variable will have its own column, so you will end with three columns of data. (Note the data set is small to ease your burden – use a two-way ANOVA regardless!) Conduct a two-way ANOVA to determine whether age and/or play group size affects empathy.

4 y/o 8 y/o

9. Paste the relevant SPSS output, including post hocs if necessary.  (2 pts)

10. Create the most appropriate graph(s) for this data given the results (make sure to have axes titles and error bars). (2 pts)

11. Write an APA-style Results section based on your analysis. All homework “Results sections” should follow the examples provided in the presentations and textbooks. They should include the statistical statement within a complete sentence that mentions the type of test conducted, whether the test was significant, and if relevant, effect size and/or post hoc analyses. Don’t forget to include a decision about the null hypothesis. (4 pts)

In what ways do frequency distributions for qualitative data differ from those for quantitative data?

4. You later learn that people from the ‘class’ group from Q3 had classmates who had also tried Duolingo. (IE these classmates had started in a ‘class only’ group, and then added ‘duolingo’). The data for these people are provided below. 4a. State the null and alternative hypotheses (1 point) 4b. Calculate the test statistic (3 points); and state your conclusion (use the same alpha from Q3) (1 point) 4c. What are some reasons why your conclusions from Q3 and Q4 differ from one another? (Especially despite the larger sample size in Q3)

Locate the results of a recent survey that shows at least two variables in a newspaper, magazine, or Internet article. Outline the survey data so that your peers can understand the variables and results, and then identify at least one key formula from this module that you could use to evaluate the data. Provide a brief explanation of why you selected the formula you did and why it matters. Also, explain what the formula is, where it is in the textbook, and clearly define your parameters. Be sure to support your statements with logic and argument, citing any sources referenced. Post your initial response early, and check back often to continue the discussion. Be sure to respond to your peers and instructors posts, as well.

Consider a standard deck of 52 cards. You select one card at random and do not replace it. You shuffle the deck and then pick another card. If you are trying to find the probability that you pick a Red card first (denoted by R) and then a King card (denoted by K), which of the following probability statements is applicable for finding the probability?

P(R )P(K|R)

n(R AND K)/n(S)

P(R|K )P(K|R)

P(R )+P(K)-P(R n K)

2. School administrators are concerned that important university emails are being automatically (and incorrectly) labeled as ‘spam’ by email providers. Independent random samples of ‘google mail’ and ‘yahoo.ca’ users are randomly selected. Out of the sample of google mail users (n=200), 150 reported that they had received the email. Out of the sample of yahoo.ca users (n=150), 90 reported that they had received the email. Is there a significant difference in receiving the email between users of google mail and yahoo.ca email providers?

2a. Calculate the test statistic.

2b. Find the p-value for the test. Test for a significant difference at the 1% significance level and state whether you reject or fail to reject the null (and why).

A sociologist is studying the age of the population in Blue Valley. Ten years ago, the population was such that 20% were under 20 years old, 14% were in the 20- to 35-year-old bracket, 34% were between 36 and 50, 24% were between 51 and 65, and 8% were over 65. A study done this year used a random sample of 210 residents. This sample is given below. At the 0.01 level of significance, has the age distribution of the population of Blue Valley changed? Under 20 20 - 35 36 - 50 51 - 65 Over 65 29 28 66 68 19 (i) Give the value of the level of significance. State the null and alternate hypotheses. H0: Ages under 20 years old, 20- to 35-year-old, between 36 and 50, between 51 and 65, and over 65 are independent. H1: Ages under 20 years old, 20- to 35-year-old, between 36 and 50, between 51 and 65, and over 65 are not independent. H0: Time ten years ago and today are independent. H1: Time ten years ago and today are not independent. H0: The population 10 years ago and the population today are independent. H1: The population 10 years ago and the population today are not independent. H0: The distributions for the population 10 years ago and the population today are the same. H1: The distributions for the population 10 years ago and the population today are different. (ii) Find the sample test statistic. (Round your answer to two decimal places.) (iii) Find or estimate the P-value of the sample test statistic. P-value > 0.100 0.050 < P-value < 0.100 0.025 < P-value < 0.050 0.010 < P-value < 0.025 0.005 < P-value < 0.010 P-value < 0.005 (iv) Conclude the test. Since the P-value ≥ α, we reject the null hypothesis. Since the P-value < α, we do not reject the null hypothesis. Since the P-value < α, we reject the null hypothesis. Since the P-value ≥ α, we do not reject the null hypothesis. (v) Interpret the conclusion in the context of the application. At the 1% level of significance, there is insufficient evidence to claim that the age distribution of the population of Blue Valley has changed. At the 1% level of significance, there is sufficient evidence to claim that the age distribution of the population of Blue Valley has changed.

Question 5 of 8 The Wechsler Adult Intelligence Scale (IQ test) is constructed so that Full Scale IQ scores follow a normal distribution, with a mean of 100, and a standard deviation of 15. Dr.Smartyskirt is a University professor and believes that university professors are smarter than the national average and wants to use it (the intelligence of the professors) as a marketing tool to bring new students to the University. A researcher is hired to conduct a study to determine whether University professors, on average, have higher Full Scale IQs than the population. A random sample of 100 professors from various Universities were given the IQ test and were found to have an average Full Scale IQ of 140. Which hypothesis test should be used to determine whether the mean Full Scale IQ score of the professors is higher than the national average? z-test for the population mean, t-test for the population mean, z-test for the population proportion, t-test for the population proportion

Question 6 of 8 What are the null and alternative hypotheses? H0: μ = 100 Ha: μ > 100 H0: μ = 100 Ha: μ ≠ 100 H0: μ = 140 Ha: μ > 140 H0: μ = 140 Ha: μ ≠ 140

Question 7 of 8 What is the value of the test statistic used to determine whether the mean Full Scale IQ score of the University professors is higher than the national average? 26.67 -26.67 2.67 -2.67

Question 8 of 8 After analyzing the data to determine whether the mean Full Scale IQ score of the University professors is higher than the national average, the P-value of < .00001 was obtained. Using a .05 significance level, what conclusion can be drawn from the data?

Reject the null hypothesis. The average Full Scale IQ of the University professors is higher than the population average. Do not reject the null hypothesis. The average Full Scale IQ of the University professors is not higher than the population average.

Do not reject the null hypothesis. The average Full Scale IQ of the University professors is higher than the population average.

Reject the null hypothesis. The average Full Scale IQ of the University professors is not higher than the population average.

1)

Assume that the national average score on a standardized test is 1010, and the standard deviation is 200, where scores are normally distributed. What is the probability that a test taker scores at least 1600 on the test?

• Round your answer to two decimal places.

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2)

Assume that the salaries of elementary school teachers in a particular country are normally distributed with a mean of $38,000 and a standard deviation of $4,000. What is the cutoff salary for teachers in the top 7%?

• Use Excel, and round your answer to the nearest dollar.

___________________________________________

3) A cookie manufacturer sells boxes of cookies that claim to weigh 16 ounces on the packaging. Due to variation in the manufacturing process, the weight of the manufactured boxes follows a normal distribution with a mean of 16 ounces and a standard deviation of 0.25 ounce. The manufacturer decides it does not want to sell any boxes with weights below the 1st percentile so as to avoid negative customer responses. What is the minimum acceptable weight, in ounces, of a box of cookies? Round your answer to two decimal places.

_____________________________________________

4) The weights of bags of raisins are normally distributed with a mean of 175 grams and a standard deviation of 10 grams. Bags in the upper 4.5% are too heavy and must be repackaged. Also, bags in the lower 5% do not meet the minimum weight requirement and must be repackaged. What are the ranges of weights for raisin bags that need to be repackaged? Use a TI-83, TI-83 plus, or TI-84 calculator, and round your answers to the nearest integer.

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5)

The resistance of a strain gauge is normally distributed with a mean of 100 ohms and a standard deviation of 0.3 ohms. To meet the specification, the resistance must be within the range 100±0.7 ohms. What proportion of gauges is acceptable?

• Round your answer to four decimal places.

________________________________________________

6)

Suppose that the weight of sweet cherries is normally distributed with mean μ=6 ounces and standard deviation σ=1.4 ounces. What proportion of sweet cherries weigh more than 4.7 ounces?

• Round your answer to four decimal places.

___________________________________________

7)

Suppose X∼N(8,1.5), and x=5. Find and interpret the z-score of the standardized normal random variable.

The z-score when x=5 is ___. The mean is ____

This z-score tells you that x=5 is ___ standard deviations to the left of the mean.

____________________________________

Question 1 of 8 According to recent statistics, the average Facebook user has 225 Facebook friends. For a statistics project a student at San Jose State College tests the hypothesis that SJSU students will average more than 225 Facebook friends. She randomly selects 3 classes from the schedule of classes and distributes a survey in these classes. Her sample contains 45 students. Here are the null and alternative hypotheses for her study: H0: µ = 255, Ha: µ > 255. What does µ represent in these hypotheses?

-Even though 240.2 is larger than 225, it is not significantly larger than 225. In other words, the data do not provide enough evidence to conclude that the mean number of Facebook friends of all SJSU college students is higher than 225.

-Nothing. The conditions for use of a t-test were not met. She cannot trust that the P-value is accurate for this reason.

-240.2 is significantly larger than 225. In other words the data provide provide enough evidence to conclude that the mean number of Facebook friends of all SJSU college students is higher than 225.

Question 2 of 8 From her survey data, the statistics student calculates that the mean number of Facebook friends for her sample is 240.2 with a standard deviation of 79.3. She analyzed her data using a t-test and obtained a P-value of p = .10
What conclusion can she draw from her data?

-Even though 240.2 is larger than 225, it is not significantly larger than 225. In other words, the data do not provide enough evidence to conclude that the mean number of Facebook friends of all SJSU college students is higher than 225.

-Nothing. The conditions for use of a t-test were not met. She cannot trust that the P-value is accurate for this reason.

-240.2 is significantly larger than 225. In other words the data provide provide enough evidence to conclude that the mean number of Facebook friends of all SJSU college students is higher than 225.

Question 3 of 8 According to Facebook’s self-reported statistics, the average Facebook user is connected to 80 community pages, groups, and events. For a statistics project a student at San Jose State University tests the hypothesis that SJSU students will average less than 80 such connections.
She posts a survey on her Facebook page. Her sample contains 45 responses.
She chooses a 5% level of significance. From her data she calculates a t-test statistic of approximately -2.14 with a P-value of about 0.019. What can she conclude?

-Nothing. The conditions for use of a T-model are not met. She cannot trust that the P-value is accurate for this reason.

-The data is not statistically significant. In other words, the data do not provide enough evidence to conclude that the mean number of Facebook connections to community pages, groups and events of all SJSU college students is less the 80.

-The data is statistically significant. In other words, the data do provide enough evidence to conclude that the mean number of Facebook connections to community pages, groups and events of all all SJSU college students is less the 80

Question 4 of 8 The average US woman wears 515 chemicals on an average day from her makeup and toiletries. A random sample from California found that on average the California woman wears 325 chemicals per day with a standard deviation of 90.5.
Which hypothesis test should be used to determine whether the sample contains less than the US average of wearing 515 chemicals per day?

- t-test for the population mean

- z-test for the population mean

- z-test for the population proportion

- t-test for the population proportion

What is meant by the term significance? How is significance level related to what you learned in previous chapters about probability? Please provide an example using significance in your work or life.

What is the main use of a one sample Z-Test? Please provide an example.

A photographer is interested in whether there is a difference in the proportions of adults and children who like to take pictures. In a survey of 310 randomly selected adults and 270 randomly selected children, 219 adults and 215 children said they like to take pictures.

a) Define Population 1 and Population 2.

b) Define the parameter of interest.

c) Name the distribution required to calculate confidence intervals. (Check the relevant criteria.)

d) Construct a 90% confidence interval for the true difference in proportions.

e) Interpret your confidence interval.

f) Is there a difference in the proportions (at a 90% CL)?

A professor states that in the United States the proportion of college students who own iPhones is .66. She then splits the class into two groups: Group 1 with students whose last name begins with A-K and Group 2 with students whose last name begins with L-Z. She then asks each group to count how many in that group own iPhones and to calculate the group proportion of iPhone ownership. For Group 1 the proportion is p1 and for Group 2 the proportion is p2. To calculate the proportion you take the number of iPhone owners and divide by the total number of students in the group. You will get a number between 0 and 1. What would you expect p1 and p2 to be? Do you expect either of these proportions to be vastly different from the population proportion of .66? Would you be surprised if p1 was different than p2? Would you be surprised if they were the same or similar? What statistical concept describes the relationship between the first letter of someone's last name and whether or not they own an iPhone?

In the following problem, check that it is appropriate to use the normal approximation to the binomial. Then use the normal distribution to estimate the requested probabilities.

Do you try to pad an insurance claim to cover your deductible? About 36% of all U.S. adults will try to pad their insurance claims! Suppose that you are the director of an insurance adjustment office. Your office has just received 140 insurance claims to be processed in the next few days. Find the following probabilities. (Round your answers to four decimal places.)

(a) half or more of the claims have been padded


(b) fewer than 45 of the claims have been padded


(c) from 40 to 64 of the claims have been padded


(d) more than 80 of the claims have not been padded

The probability of winning on a lot machine is 5%. If a person plays the machine 500 times, find the probability of winning 30 times. Use the normal approximation to the binomial distribution.

A travel survey of 1500 Americans reported an average of 7.5 nights stayed when they went on vacation. Find a point estimate of the population mean. If we can assume the population standard deviation is 0.8 night, find the 95% confidence interval for the true mean.

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