STATS 101/108 - Past exam S123

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STATS 101/108 - Past exam S123

Question 1

A random sample of 40 photos of dogs was obtained from Unsplash.com. For each photo the number of likes per 100 000 views was calculated.

The sample data was used to generate a VIT bootstrap confidence interval to estimate the proportion of all dog photos on Unsplash.com with more than 10 likes per 100 000 views.

Q1a

Use the information above to complete the following statements:

A proportion of 54% one of the plausible values provided by the VIT bootstrap confidence interval for the proportion of all dog photos on Unsplash.com with more than 10 likes per 100 000 views.

Because of the plausible values in this confidence interval are greater than 50%, this confidence interval the claim that “most dog photos on Unsplash.com have more than 10 likes per 100 000 views”.

The population referred to in the interpretation the same as the population the sample was taken from.

Q1b

Use the information above to decide if each of the following three statements are TRUE or FALSE.

If the sample size had been 80, rather than 40, the confidence interval would be wider than the one shown above

If the sample proportion had been 80%, rather than 70%, the confidence interval would be narrower than the one shown above

If the sample data was used to estimate the proportion of all dog photos on Unsplash.com that do not have more than 10 likes per 100 000 views, then the confidence interval would have a similar width to the one shown above

Q1c

Using the random sample of 40 dog photos from Unsplash.com, bootstrapping was used to generate a confidence interval for the median number of likes per 100 000 views.

The VIT output is below:

Below are two research questions.

Research Question 1: For all photos of dogs on Unsplash.com, what is the median number of views per 100 000 likes?

Research Question 2: For these 40 photos of dogs from Unsplash.com, what is the median number of views per 100 000 likes?

Choose the research question that requires a sample-to-population inference and answer it by writing a one sentence interpretation of the confidence interval.

Question 2

An online Artificial Intelligence (AI) tool can analyse features of photos of people and provide an “attractiveness rating” on a scale from 1 (not attractive) to 9 (extremely attractive).

A new random sample of 40 photos of people was taken from Unsplash.com. Each photo was rated for attractiveness using this AI tool.

Q2a

Read the short article excerpt below:

A startup company called Qoves has developed an AI-powered beauty assessment tool that tells you how attractive you are. The free version spits out a list of your “predicted flaws'' and explains what sort of surgical interventions and expensive serums are needed to “fix” you. They’re not the only ones playing around with AI for this purpose, there are an alarming number of these sort of beauty assessment algorithms online. Face++, a facial recognition platform developed by China’s Megvii Technology, for example, has a tool that gives you a “Beauty Score”. It also purports to tell you how attractive you are from both a male and female perspective.

How does a beauty scoring algorithm work? Good question; nobody really knows. Face++ won’t reveal the workings of its algorithm. Qoves’s tool, meanwhile, is trained on a dataset of hundreds of thousands of pictures scored manually by humans and extrapolates attractiveness from that. One thing we can say for certain about these sorts of tools is that they often reflect Eurocentric beauty biases. 2016, for example, saw the world’s first international beauty contest judged entirely by algorithm. Guess what? Out of 44 winners, nearly all were white, a handful were Asian, and only one had dark skin.

In no more than two sentences, discuss why using an AI tool to rate attractiveness, for a sample of Unsplash photos of people, may be problematic. In your response include at least two relevant things you learned from this article.

Q2b

A 95% confidence interval for the mean attractiveness rating for photos of people on Unsplash.com was generated using the confidence interval calculator. The limits of this confidence interval are (7.2, 8.3).

The margin of error for this confidence interval is.

The value of the point estimate used for the confidence interval is.

Question 3

Q3a

Another online AI tool analyses features of any image and rates how aesthetically appealing that image is.

The two samples of photos of dogs and people (40 photos of each) from Unsplash.com were all given an aesthetic rating out of 100 using this AI tool.

Ratings closer to 100 are given to photos which are more visually appealing, based on elements such as composition, colour, texture, and emotion.

Below are plots of this data.

Use the information above to answer the following questions.

The mean aesthetic rating for the sample of photos of people is approximately.

The standard deviation of the aesthetic ratings for the sample of photos of dogs is approximately.

The variability of the aesthetic ratings of the photos of dogs is the variability of the aesthetic ratings of the photos of people.

If 95% confidence intervals were constructed using each sample, the confidence interval for the mean aesthetic ratings of photos of dogs on Unsplash.com would be.

Q3b

The two samples of photos of dogs and people were then used to construct a 95% confidence interval for the difference between the mean aesthetic rating of photos of dogs on Unsplash.com and the mean aesthetic rating of photos of people on Unsplash.com.

The limits of this confidence interval are (11.1, 41.0).

Use the information above to answer the following questions.

The point estimate used for the confidence interval is a.

of the plausible differences given in the confidence interval are positive.

Therefore we claim that the mean aesthetic rating for photos of dogs on Unsplash.com is higher than the mean aesthetic rating for photos of people on Unsplash.com.

A different random sample of photos of dogs and photos of people was collected. This time a 95% confidence interval constructed from the data collected had limits (10.3, 41.8).

This new confidence interval is than the first confidence interval.

Q3c

A photographer wanted to find out how visually appealing people found their photographs and whether certain compositions were more aesthetically pleasing than others. To do this, they posted a link to a survey on his Facebook page. Anyone who viewed the Facebook post could then go and complete the survey and their responses would be anonymous. 200 people completed the survey.

Use the information above to decide each of the following three statements are TRUE or FALSE.

Selection is a potential source of bias in this survey as the respondents selected themselves to take part in the survey

The sample size is a potential source of bias in this study

A 95% confidence interval accounts for all of the sources of uncertainty that might weaken the strength of our inference

Question 4

A study investigated how profile pictures of women can influence how they are perceived. The study participants were 58 adolescent girls and 60 young adult women. Each participant was randomly allocated to view a Facebook profile with either a sexualised profile photo or a non-sexualised profile photo, after which they evaluated the profile owner by answering questions.

The following variables were collected:

photo_type - the photo type viewed by the participant: sexualised, non-sexualised

age - the age group the participant was from: adolescent, young adult

competence - the participant’s rating (scale from 1, strongly agree to 7, strongly disagree) of the woman in the profile regarding their level of agreement with the statement “I have confidence in her ability to get the job done”

attractiveness - the participant’s rating (scale from 1, strongly agree to 7, strongly disagree) of the woman in the profile regarding their level of agreement with the statement “I think she is quite pretty”

Q4a

Researcher A is interested in whether there is an association between age and attractiveness .

Analysis of the variable age would involve.

Analysis of the variable attractiveness would involve.

A suitable research question involving these two variables would be: Is there an association between the age of the participant and the attractiveness rating they give?

A two-sided alternative hypothesis based on these two variables could be “The underlying attractiveness rating given by adolescent girls is that given by young adult women”.

Q4b

Researcher A carried out a two sample t-test to test for a difference between the underlying mean attractiveness rating given by adolescent girls and the underlying mean attractiveness rating given by young adult women.

The iNZight Lite output from this test is below.

Which one of the following two sentence summaries, based on the output above, is correct?

Summary 1: We have no evidence that adolescent girls give, on average, different attractiveness ratings (p = 0.1539) compared to young adult women. The underlying mean attractiveness rating given by adolescent girls is somewhere between 0.63 lower and 0.099 higher than that given by young adult women (95% confidence).

Summary 2: We have evidence that adolescent girls give, on average, the same attractiveness ratings (p = 0.1539). With 95% confidence, the underlying mean attractiveness rating given by adolescent girls is somewhere between 0.63 lower and 0.099 higher than that given by young adults.

Q4c

Researcher B is interested in whether there is a link between photo_type and competence .

A two sample t-test was carried out in iNZight Lite to test for a difference between the underlying mean competence rating for the two photo types.

The iNZight Lite output from this test is below.

Note that the lower and upper limits of the confidence interval have been deliberately hidden.

Use the information to decide if each of the following three statements are TRUE or FALSE.

The observed mean competence rating for the nonsexualised photo is higher than that for the sexualised photo.

The p-value for this test indicates that the difference between the two means is statistically significant at the 5% level.

Based on the size of the p-value and the observed difference, the 95% confidence interval for the difference between the underlying means contains positive and negative values.

Q4d

Recall that:

Researcher A was interested in whether there is an association between age and attractiveness . Researcher A conducted a two sample t-test and found a difference of 0.26 between the mean attractiveness ratings of the two age groups (p-value = 0.1539), with the adolescents having the lower mean.

Researcher B was interested in whether there is a link between photo_type and competence . Researcher B conducted a two sample t-test and found a difference of 0.96 between the mean competence ratings of the two photo types (p-value < 0.0001), with the sexualised photo having the lower mean.

Use the information to decide if each of the following three statements are TRUE or FALSE.

The type of conclusion that Researcher A could make is a causal claim.

An appropriate scope of inference for any conclusion made by Researcher A is “people like those who participated in the study”.

Based on the design of the study conducted, the test carried out by Researcher B and the p-value reported (using a 5% level of significance), a claim could be made that the sexualised photo caused a lower average competence rating for these participants.

Q4e

Researcher C was interested in whether there is a relationship between age and competence .

The visualisation below shows a T distribution with df = 28 and represents the calculation of a p-value for a two sample t-test, using an appropriate test statistic.

Use the distribution shown above to decide if each of the following statements are TRUE or FALSE.

The probability represented as the dark green area shaded under the curve corresponds to the probability statement pr(T < 0.72)

The dark green shaded area under the curve represents a probability of approximately 0.24, rather than 0.48.

The visualisation represents the calculation of a p-value for a two-sided test.

Question 5

A study investigated photos of Chief Executive Officers (CEOs). The researchers used photos of people in three different roles: CEOs from small companies, photos of CEOs from large companies, and photos of non-CEOs.

All photos were shown to a panel of participants who were asked to rate, on a scale of 0 to 100, how competent they believed the person to be.

An ANOVA (F-test) was carried out to explore the relationship between role and competence . Some of the iNZight Lite output from this test is shown below:

Q5a

Use the information above to answer the following questions:

Which group has the highest sample mean for competence ?

What is the ratio of the largest sample standard deviation to the smallest standard deviation for the groups? (round your answer to one decimal place)

The p-value for an ANOVA F-test using this data is 0.0134. This makes sense as of the pairwise differences have p-values less than 0.05.

At the 5% level of significance, there evidence that the underlying mean competency rating given to the photos depends on role.

Q5b

Use the information provided to decide if each of the following three statements are TRUE or FALSE.

We can claim that the underlying mean competency rating for large company CEOs is higher than the underlying mean competency rating for small company CEOs.

It is plausible that the underlying mean competence rating for small company CEOs is the same as the underlying mean competence rating for non-CEOs.

With 95% confidence, the underlying mean competence rating for large company CEOs is between 1.6 points lower and 16.0 points higher than the underlying mean competence rating for non-CEOs.

Question 6

A study investigated whether different photos were more or less suitable for different contexts. Student volunteers participated in the study.

Each volunteer was randomly assigned one of the scenarios from the list below:

applying for a high-salary consultant position and selecting a photo for a resume

uploading a photo to an online-dating Web site

uploading a new Facebook photo

running for the office of town mayor and selecting a photo for campaign posters

auditioning for the role of the villain in an upcoming film and selecting a photo for the application

Each participant viewed the same five photos of the same individual, such as the example below, and was asked to choose which photo they would use for the scenario they had been assigned (consultant, dating, facebook, mayor, villain).

The researchers carried out a Chi-square test for independence, using scenario as the explanatory variable and image_chosen as the response variable.

Below is some of the iNZight Lite output from their analysis:

Q6a

Use the information above to decide if each of the following three statements are TRUE or FALSE.

A suitable null hypothesis for this test would be: The underlying distribution of image chosen does not depend on the scenario.

A suitable alternative hypothesis for this test would be: The underlying distribution of image chosen is different for each scenario.

When conducting this test using iNZight Lite, image_chosen would be selected as the first variable.

Q6b

Use the information above to complete the following statements.

According to this data, how many times as likely was someone to choose image 1 compared to image 2 for the mayor scenario? (round your answer to one decimal place).

Under the null, what proportion of the images chosen for the dating scenario would be expected to be image 3? % (round to 1 dp)

Q6c

The biggest difference between the proportion of participants selecting image 1 versus image 2 is for the scenario.

The chi-square-test statistic is relatively , indicating a relatively discrepancy between what we see in the data and what we would expect to see if the null hypothesis were true.

At the 1% level of significance, there evidence that the image chosen is dependent on the scenario.

Q6d

Based on this study, write a one sentence conclusion that interprets the p-value for the Chi-square test shown in the output above. Then write one additional sentence that clearly describes what type of conclusion you are able to make and your scope of inference.