Spearman’s rank (+ limitations of pearson / spearman): notes and practice questions

Dr. Anya Sharma, a university researcher, is investigating the belief that students who spend more time studying tend to achieve higher exam scores. She randomly selects eight students and collects data on their average weekly study hours and their final exam scores.

Her data is shown in Table 1.

Table 1

Student | S1 | S2 | S3 | S4 | S5 | S6 | S7 | S8

---|---|---|---|---|---|---|---

Average weekly study hours | 10 | 15 | 5 | 20 | 12 | 8 | 25 | 18

Final exam score (out of 100) | 70 | 60 | 55 | 90 | 80 | 65 | 95 | 75

Dr. Sharma decides to calculate the Spearman's rank correlation coefficient.

Complete the table of ranks shown in Table 2, assigning rank 1 to the highest value.

Table 2

Student | S1 | S2 | S3 | S4 | S5 | S6 | S7 | S8

---|---|---|---|---|---|---|---

Rank – Study Hours | 6 | 4 | 8 | 2 | 5 | 7 | 1 | 3

Rank – Exam Scores | | | | | | | |

Calculate the value of $r_s$, Spearman's rank correlation coefficient.

Dr. Sharma believes that students with a higher number of weekly study hours achieve higher exam scores. She carries out a hypothesis test using a 10% significance level with the following null hypothesis:

$H_0$: In the population, there is no monotonic relationship between the number of weekly study hours and final exam scores.

Write down Dr. Anya Sharma's alternative hypothesis.

The critical value of $r_c$ for this test is 0.643.

State the conclusion of the hypothesis test, giving a reason.

Dr. Anya Sharma, a sports scientist, is investigating the relationship between training habits and performance in junior athletes. She wants to collect data on the weekly training hours of junior swimmers. She decides to interview every 5th swimmer entering the training facility until she has a sample of 50 swimmers.

State the sampling method Dr. Sharma has used.

Dr. Sharma constructed the following box and whisker diagram to show the weekly training hours (in hours) of a sample of junior swimmers.

Write down the median weekly training hours.

Calculate the interquartile range for the weekly training hours.

One swimmer in the sample reported training for 15 hours per week. Dr. Sharma believes this swimmer's training time is not an outlier.

Determine whether Dr. Sharma is correct. Support your reasoning.

Dr. Sharma also collected data on the average weekly training hours ($x$) and the competition score ($y$) for a group of athletes. These data are represented on the scatter diagram.

Describe the correlation between weekly training hours and competition score.

Dr. Sharma correctly calculates the equation of the regression line $y$ on $x$ for these athletes to be $y = -2.5x + 110$. She uses the equation to estimate the competition score for an athlete who trains 3 hours per week.

Find the competition score calculated by Dr. Sharma.

State whether it is valid to use the regression line $y$ on $x$ for Dr. Sharma's estimate in part (f). Give a reason for your answer, assuming the original data for training hours ranged from 5 to 20 hours.

Dr. Sharma investigated the relationship between an athlete's national competition rank and their average daily protein intake (in grams). She collected data for eight athletes, as shown in the table.

Dr. Sharma intends to analyse the data using Spearman's rank correlation coefficient, $r_s$.

Copy and complete the information in the following table.

Calculate the value of $r_s$.

Interpret your result.

Ms. Chen, a small business owner, wants to investigate if there is a monotonic relationship between the monthly advertising budget and the number of units of a new product sold. She collects data for eight months, as shown in Table 1.

Table 1: Monthly Advertising Budget and Units Sold

Ms. Chen decides to calculate the Spearman's rank correlation coefficient. Complete the table of ranks shown in Table 2.

Table 2: Ranks for Advertising Budget and Units Sold

Calculate the value of $r_s$, Spearman's rank correlation coefficient.

Ms. Chen believes that a higher advertising budget leads to more units sold. She carries out a hypothesis test using a 10% significance level with the following null hypothesis:

$H_0$: In the population, there is no monotonic relationship between the monthly advertising budget and the number of units sold.

Write down Ms. Chen's alternative hypothesis.

The critical value of $r_c$ for this test is 0.643.

State the conclusion of the hypothesis test, giving a reason.

(a) TechInnovate is considering collecting more data for their analysis.

(i) State one advantage of increasing the sample size.

(ii) State one disadvantage of increasing the sample size.

(b) The production manager at Plant Alpha recorded the time, in minutes, taken to produce a batch of electronic components for 10 randomly selected batches:

$18.2, 19.5, 17.8, 20.1, 18.5, 19.0, 17.5, 20.5, 18.8, 19.3$

Find the value of $s_{n-1}$ for this sample from Plant Alpha.

(c) A manager claims that Plant Alpha's production times are more consistent than Plant Beta's. Given that the sample standard deviation ($s_{n-1}$) for Plant Beta's production times is $1.05$ minutes, make one criticism of this claim.

(d) TechInnovate wants to compare the mean production times of Plant Alpha and Plant Beta using a pooled t-test.

(i) State the condition regarding population variances required to use a pooled t-test.

(ii) Given that for Plant Beta, a sample of $12$ batches yielded a mean production time of $\bar{x}_B = 19.3$ minutes and a sample standard deviation of $s_B = 1.05$ minutes, state whether TechInnovate should use a pooled t-test in this case. Justify your answer.

(e) TechInnovate believes Plant Alpha has a lower mean production time than Plant Beta.

(i) State appropriate null and alternative hypotheses for the pooled t-test.

(ii) Find the p-value.

(iii) Given that the test is carried out at the 5% significance level, state the appropriate conclusion in context. Justify your answer.

(f) The company also investigates the relationship between operator experience (in years) and the number of defective items produced per day. A sample of 8 operators yielded the following data:

(i) Assuming all requirements are met, perform a test at the 5% significance level to determine if there is a linear correlation between operator experience and the number of defective items. State the hypotheses and justify your conclusion.

(ii) If the requirements for this test are not met, state an alternative test that could be used.

(g) For the data in (f.i), the equation of the least squares regression line of defective items ($D$) on operator experience ($E$) is $D = -1.5E + 18.25$. Give, in context, an interpretation of the gradient $-1.5$ in this model.

(h) TechInnovate uses a baseline model to predict the number of defective items ($D_{pred}$) for a batch based on its size ($S$): $D_{pred} = 0.5S + 10$. The "Quality Deviation" ($Q$) for a batch is defined as $Q = D_{pred} - D_{actual}$. A positive Quality Deviation indicates better-than-expected quality.

(i) Show that for a batch of $150$ components from Plant Beta that produced $80$ defective items, the Quality Deviation is $5.0$.

(ii) To compare quality control, samples of Quality Deviation scores were collected:

• Plant Alpha: $n_{QA} = 15$, $\bar{x}_{QA} = 4.5$, $s_{QA} = 1.2$

• Plant Beta: $n_{QB} = 18$, $\bar{x}_{QB} = 3.8$, $s_{QB} = 1.1$

Assuming that the appropriate requirements are met, use a pooled t-test at a 5% significance level to determine if the mean Quality Deviation is higher in Plant Alpha than in Plant Beta. Write down your null and alternative hypotheses and justify your conclusion.

(i) Using the results from parts (e) and (h.ii), explain how each plant could claim they are performing better than the other plant.

A fitness coach wanted to investigate if there was a relationship between the number of hours clients spent in a specific yoga class per week and their flexibility score (measured on a scale of 1 to 100). The data from $10$ clients is shown in the table below.

Calculate Spearman's rank correlation coefficient ($r_S$) for this data.

Interpret the value of $r_S$ and comment on its validity.

A health researcher is investigating the relationship between the number of hours individuals spend exercising per week and their average weekly stress level (rated on a scale of $1$ to $10$, where $10$ is very high stress). The data for $10$ participants is shown below:

(a) Construct a table of ranks for this data.

(b) Calculate Spearman's rank correlation coefficient for this data.

(c) The researcher concludes that increased exercise leads to lower stress levels. Using your calculations, comment on whether or not the researcher's conclusion is supported by the data. Suggest what conclusions you are able to make from your calculations.

(a) A human resources manager collected data on the 'Employee Rank' (an ordinal measure assigned by senior management) and the average 'Customer Satisfaction Rating' (on a scale of $0$ to $100$) for eight employees. The data is shown in the table below:

Explain why it might not be appropriate to use the Pearson's product-moment correlation coefficient (PMCC) in this case.

(b) Calculate Spearman's rank correlation coefficient ($r_S$) for this data.

(c) Interpret the value of $r_S$ and comment on its validity.

Two music critics, Liam and Chloe, independently rank eight newly released albums from 'Album 1' to 'Album 8' based on their artistic merit.

The albums are labelled 1 to 8 and the critics' ranks are shown in the table.

(a) Write down the rank that Liam awards Album 3.

(b) Calculate Spearman's rank correlation coefficient for these data.

(c) Comment on your answer to part (b) in terms of the ranks awarded by Liam and Chloe.

The scores of eight students in a national mathematics competition and the number of hours they spent studying are shown in the following table.

(a)(i) For this data, find the upper quartile.

(a)(ii) For this data, find the interquartile range.

(b) Determine if Student A's score is an outlier for this data. Justify your answer.

A researcher is investigating the relationship between students' mathematics competition scores and their study hours to determine whether study hours can reasonably be used to predict a student's score.

The study hours of the students are shown in the table.

The researcher finds that, for this data, the Pearson's product moment correlation coefficient is $r = 0.35$.

(c) State whether it would be appropriate for the researcher to use the equation of a regression line for $y$ on $x$ to predict a student's score. Justify your answer.

The researcher then decides to find the Spearman's rank correlation coefficient for this data, and creates a table of ranks (lowest value = rank 1).

(d)(i) Write down the value of:

a,

(d)(ii) Write down the value of:

b,

(d)(iii) Write down the value of:

c.

(e)(i) Find the value of the Spearman's rank correlation coefficient $r_s$.

(e)(ii) Interpret the value obtained for $r_s$.

(f) When calculating the ranks, the researcher incorrectly read Student A's score as 90. Explain why the value of the Spearman's rank correlation $r_s$ does not change despite this error.

A botanist is investigating the relationship between the concentration of a new liquid fertilizer and the average height of a particular plant species. They prepare seven different concentrations of the fertilizer and grow a sample of plants at each concentration, recording the average height after a specific period.

The data collected is shown in the table below:

Write down the value of the Spearman's rank correlation coefficient, $r_s$.

Find the Pearson's product-moment correlation coefficient, $r$.

Use your value of $r$ to state which two of the following would best describe the correlation between fertilizer concentration and average plant height.

Positive Negative Strong Weak No correlation

The relationship between fertilizer concentration and average plant height can be modelled by the regression equation $y = ax + b$.

Write down the value of $a$.

Write down the value of $b$.

According to this model, state in context what the value of $b$ represents.

A botanist uses the regression equation to estimate the average height of a plant grown with a fertilizer concentration of $2.8$ g/L.

Find this estimated height.

State two reasons that the botanist might use to justify the validity of this estimate.

To investigate the effectiveness of different fertilizer brands, the botanist conducts an experiment. They grow two groups of plants, one using 'Bio-Grow' (Brand A) and another using 'RootBoost' (Brand B), both at a standard concentration. They take a random sample of seven plants from each group and record their average heights (in cm) after a month.

Brand A (Bio-Grow) heights: $14.2, 15.1, 13.9, 14.8, 15.5, 14.5, 15.3$

Brand B (RootBoost) heights: $16.1, 17.0, 15.8, 16.5, 17.2, 16.3, 16.9$

The botanist conducts a t-test, at the $5\%$ level of significance, to see if the mean plant height using Brand A is different from the mean plant height using Brand B. They assume the population variances are the same.

For this test, the null hypothesis is $\mu_A = \mu_B$.

Write down the alternative hypothesis.

Find the $p$-value for this test.

State the conclusion of the test. Justify your answer.

State one additional assumption the botanist has made about the distributions to conduct this test.

A consumer electronics magazine conducted a review of 10 new smart home devices. Each device was rated on two criteria: a "User Satisfaction Score" (out of 10, based on extensive user feedback) and an "Expert Review Score" (out of 5, by professional reviewers).

The data for the 10 devices is shown in the table below:

(a) For the User Satisfaction Score ($X$),

(i) find the upper quartile.

(ii) find the interquartile range.

(b) Show that the highest User Satisfaction Score is not an outlier for this data.

The scores for both criteria were ranked to calculate Spearman's rank correlation coefficient, $r_s$. The ranks for the Expert Review Score ($Y$) are shown in the following table:

(c) Write down the value of

(i) a

(ii) b

(iii) c.

(d) (i) Find $r_s$.

(ii) If P2's Expert Review Score is upgraded from $3.5$ to $3.6$, explain why the value of $r_s$ does not change.

The magazine's editor concludes from this data that devices with high User Satisfaction Scores are very likely to also receive high Expert Review Scores.

(e) State, with a reason, whether the editor's conclusion is appropriate.

(a) A school counsellor is investigating the relationship between students' performance in Mathematics and Physics. They collect data from 8 students, recording their scores (out of 100) in a recent exam for both subjects. The data is shown in the table below.

(i) Write down the value of the Pearson's product-moment correlation coefficient, $r$.

(ii) Using the value of $r$, interpret the relationship between students' Math scores and Physics scores.

(b) Write down the equation of the regression line $y$ on $x$.

(c) (i) Use your regression equation from part (b) to estimate a student's Physics score if they achieved a perfect 100 in Mathematics.

(ii) State whether this estimate is reliable. Justify your answer.

(d) The counsellor also wants to calculate the Spearman's rank correlation coefficient. Copy and complete the information in the following table by ranking the scores.

(e) (i) Find the value of the Spearman's rank correlation coefficient, $r_s$.

(ii) Comment on the result obtained for $r_s$.

(f) The counsellor later realizes there was a marking error for student S4's Physics score and adjusts it from $79.9$ to $77.0$. Explain why the value of the Spearman's rank correlation coefficient $r_s$ does not change.

Two food critics, Marcus and Elena, independently evaluate and rank eight new restaurants, labelled A to H, in a city.

The ranks awarded by the two critics are shown in the table.

Write down the rank awarded to Restaurant E by Elena.

(b) Calculate Spearman's rank correlation coefficient, $r_s$, for these data.

(c) Comment on your answer to part (b) in the context of the rankings awarded by Marcus and Elena.