Measures of central tendency / dispersion (std. Dev, var, IQR) + effect of constant changes to data: notes and practice questions

A group of ten students recorded the number of hours they spent studying for their final IB Mathematics exam in the week leading up to it.

Their results are shown below:

5, 7, 6, 8, 5, 9, 7, 6, 5, 10

For this data set, find the value of

(a) the mode.

A group of ten students recorded the number of hours they spent studying for their final IB Mathematics exam in the week leading up to it.

Their results are shown below:

5, 7, 6, 8, 5, 9, 7, 6, 5, 10

For this data set, find the value of

(b) the mean.

A group of ten students recorded the number of hours they spent studying for their final IB Mathematics exam in the week leading up to it.

Their results are shown below:

5, 7, 6, 8, 5, 9, 7, 6, 5, 10

For this data set, find the value of

(c) the standard deviation.

(a) A botanist is studying a rare species of flower. They record the number of petals ($p$) on a sample of these flowers. The data is presented in the frequency table below:

$p$

5

6

7

8

9

10

Frequency

3

8

20

15

7

2

Find an unbiased estimate of the population mean number of petals for this rare flower species.

(b) Find an unbiased estimate of the population variance of the number of petals for this rare flower species.

(c) A botanist suspects that the average number of petals for this rare species is different from the average of 7.0 petals observed in a more common, related species. She sets up a hypothesis test with the null hypothesis $H_0: \mu = 7.0$.

(i) State the alternative hypothesis.

(ii) Given that all assumptions for this test are satisfied, carry out an appropriate hypothesis test. State and justify your conclusion, using a 10% significance level.

A software development company tracked the completion times of 150 projects. The cumulative frequency graph shows the completion times obtained by the projects.

Find the median completion time of the projects.

The projects were assigned a performance tier from 1 to 5, depending on the completion time. The number of projects receiving each tier is shown in the following table.

Find an expression for $p$ in terms of $q$.

The mean performance tier for these projects is 3.5.

Find the number of projects that obtained a tier 5.

Find the minimum completion time needed to obtain a tier 5.

The following tables show the mean monthly rainfall, by month, in two different regions: Green Valley and Sunstone Coast.

Green Valley

Month | Mean monthly rainfall (mm)

---|---

January | $80$

February | $75$

March | $90$

April | $60$

May | $50$

June | $30$

July | $20$

August | $25$

September | $40$

October | $70$

November | $95$

December | $85$

Sunstone Coast

Month | Mean monthly rainfall (mm)

---|---

January | $40$

February | $35$

March | $45$

April | $50$

May | $60$

June | $70$

July | $80$

August | $75$

September | $65$

October | $55$

November | $48$

December | $42$

(a) Find the mean monthly rainfall over the course of the year for Green Valley.

(b) Find the standard deviation of the monthly rainfall in Green Valley.

(c) Find the mean monthly rainfall over the course of the year for Sunstone Coast.

(d) Find the standard deviation of the monthly rainfall in Sunstone Coast.

(e) By referring directly to your answers from parts (a) -(d), make contextual comparisons about the monthly rainfall in Green Valley and Sunstone Coast throughout the year.

A factory conducts quality control checks on the diameter (in mm) of components produced by Production Line A. A sample of measurements is recorded as:

18.2, 19.5, 20.1, 20.3, 20.5, 20.6, 20.8, 21.0, 21.2, 21.5, 22.0, 22.1, 22.3, 22.5, 23.0

For these data, the lower quartile is 20.3 mm and the upper quartile is 22.1 mm.

Show that a component with a diameter of 18.2 mm would not be considered an outlier.

Another production line, Line B, also produces similar components. The box and whisker diagram below displays the diameters (in mm) of a sample of components from Line B.

A quality control manager reviews the box and whisker diagrams for both lines and suggests that Production Line B generally produces components with larger diameters.

With reference to the box and whisker diagrams for Line A (from part (a) ) and Line B, state one aspect that may support the manager's opinion and one aspect that may counter it.

A logistics company recorded the delivery times (in minutes) for a large batch of packages. The data is grouped in the frequency table below:

Delivery Time (minutes) | Frequency

---|---

$20 \le t < 25$ | $8$

$25 \le t < 30$ | $22$

$30 \le t < 35$ | $55$

$35 \le t < 40$ | $98$

$40 \le t < 45$ | $135$

$45 \le t < 50$ | $110$

$50 \le t < 55$ | $70$

$55 \le t < 60$ | $35$

$60 \le t < 65$ | $15$

$65 \le t < 70$ | $2$

(a) Calculate estimates of the mean and standard deviation of the delivery times.

(b) Construct a cumulative frequency table for the data, and use it to draw a cumulative frequency curve.

(c) Use your graph to estimate:

(i) the median delivery time

(ii) the lower and upper quartile of the delivery times

(iii) the interquartile range

(iv) the $85$th percentile of delivery times.

(d) Draw a box-and-whisker plot of the data.

(e) Determine, with reasons, whether any customers could be considered outliers.

A group of 160 participants completed a fitness challenge. The cumulative frequency graph shows the points obtained by the participants.

Find the median of the points obtained.

The participants were awarded a performance grade from 1 to 5, depending on the points obtained in the challenge.

The number of participants receiving each grade is shown in the following table.

Find an expression for $a$ in terms of $b$.

The mean grade for these participants is 3.5.

Find the number of participants who obtained a grade 5.

Find the minimum points needed to obtain a grade 5.

A survey was conducted to investigate the daily screen time (in minutes) of 80 students. The results are presented in the frequency table below.

Time ($t$ minutes) | Frequency

---|---

$0 \le t < 30$ | $6$

$30 \le t < 60$ | $10$

$60 \le t < 90$ | $22$

$90 \le t < 120$ | $20$

$120 \le t < 150$ | $12$

$150 \le t < 180$ | $7$

$180 \le t < 210$ | $3$

(a) State the modal class.

(b) Find the class in which the median time lies.

(c) Construct a cumulative frequency table for this data.

(d) Sketch the cumulative frequency curve.

(e) Use your curve to find estimates for the median and interquartile range.

[Maximum mark: 19]

A tech company recorded the time (in minutes) 180 customers spent completing a new online feedback survey. The data was compiled into the following cumulative frequency graph.

(a) Use the graph to find

(i) the median time;

(ii) the lower quartile;

(iii) the upper quartile;

(iv) the interquartile range.

Sarah completed the survey in 1.5 minutes.

(b) Determine whether Sarah's time is an outlier.

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.

[Maximum mark: 4]

A salesperson records the number of items sold per day over several weeks. The data is presented in the following frequency distribution table.

| Frequency (f)

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

1 | 3

2 | 5

3 | 8

4 | p

5 | 6

6 | 4

)

For this distribution, the mean number of items sold per day is 3.8.

(a) Write down the total number of days the salesperson recorded data in terms of p.

(b) Calculate the value of p.

A quality control manager at a manufacturing plant wants to assess the consistency of a new batch of electronic components. He decides to test ten components, ensuring that five are selected from Production Line A and five from Production Line B. The manager instructs the supervisors of each line to provide the required number of components from their current production.

(a) Name the type of sampling that best describes the method used by the quality control manager.

The weights, in grams, of the ten components selected for the test are:

$148, 153, 161, 155, 142, 160, 150, 157, 145, 159$.

(b) For these ten components, find

(i) the mean weight.

(ii) the standard deviation of the weights.

The target weight for these components is $155$ g. The manager is concerned that the components might be consistently underweight. Perform an appropriate test at the $10\%$ significance level to see if the mean weight of the components produced is less than the target weight. It can be assumed that the weights come from a normal population.

State one reason why the test performed in part (c) might not be valid.

Two additional components are tested at a later date. The mean weight for all twelve components is $154.5$ g and the standard deviation is $7.2$ g.

For further analysis, a 'quality score' for the twelve components is obtained by multiplying the weights by $1.5$ and subtracting $50$.

(e) For the twelve components, find

(i) their mean quality score.

(ii) the standard deviation of their quality score.

A tech company launched two new smartphone models, "Voyager" and "Explorer". They collected customer satisfaction scores (out of 100) from a large sample of users for both models. The results are summarized in the following box and whisker diagram.

Identify which two of the following statements must be true according to the box and whisker diagram. Indicate your choices by placing tick marks in the second column of the following table.

Statement | True (✓)

---|---

The satisfaction scores for Model Voyager are normally distributed. |

A higher percentage of customers gave a score less than 70 for Model Voyager than for Model Explorer. |

A higher percentage of customers gave a score greater than 90 for Model Explorer than for Model Voyager. |

The interquartile range for Model Explorer is less than the interquartile range for Model Voyager. |

A product manager believes there is no significant difference in the average customer satisfaction scores between the two models. She plans to conduct a t-test at the 10% significance level. Write down the null and alternative hypotheses for her test.

The t-test yielded a p-value of 0.0783. Find the p-value for her test.

Write down the conclusion to the test. Give a reason for your answer.

The lifespans, $t$, of 250 LED light bulbs are recorded in the following table.

This table is used to create a cumulative frequency graph.

Write down the mid-interval value of the class $0 \le t < 1000$.

Calculate an estimate of the mean lifespan of the 250 light bulbs.

Use the cumulative frequency curve (which would be provided in an exam) to estimate the interquartile range. Assume the lower quartile ($Q_1$) is $1300$ hours and the upper quartile ($Q_3$) is $2150$ hours.

A light bulb from the data set had a lifespan of $3400$ hours.

Use your answer to part (b) to estimate whether this light bulb's lifespan is an outlier for this data. Justify your answer.

It is believed that the lifespans of these LED light bulbs follow a normal distribution with mean $1740$ hours and standard deviation $450$ hours.

It is decided to perform a $\chi^2$ goodness of fit test on the data to determine whether this sample of 250 light bulbs could have plausibly been drawn from an underlying distribution $N(1740, 450^2)$.

Write down the null and the alternative hypotheses for the test.

As part of the test, the following table is created.

Find the value of $a$ and the value of $b$. Give your answers to one decimal place.

Hence, perform the test to a 5% significance level, clearly stating the conclusion in context.

A logistics company recorded the weights, in kilograms (kg), of six parcels dispatched in a single hour:

1.25 kg, 1.80 kg, 1.50 kg, 1.25 kg, 2.10 kg, 1.60 kg

For these six parcels, find

(i) the mean weight.

(ii) the median weight.

(iii) the modal weight.

(iv) the range of the weights.

A new parcel is added to the shipment. Its weight is measured as 1.75 kg to the nearest 10 grams.

Write down the shortest possible weight of this new parcel.

A company manufactures specialized medical devices. Each device consists of a main circuit board and a protective casing. The weight of the circuit board, $C$, is normally distributed with a mean of $120$ g and a standard deviation of $5$ g. The weight of the protective casing, $P$, is normally distributed with a mean of $30$ g and a standard deviation of $2$ g. The weights of the circuit board and the casing are independent.

Find the probability that a randomly chosen complete device has a total weight of less than $145$ g.

A batch of $10$ such devices is to be packed into a container. The container has a maximum weight capacity of $1510$ g. The weight of each device is independent.

Find the probability that the total weight of the $10$ devices is greater than the capacity of the container.

The company sources a critical microchip component from two different suppliers, Supplier X and Supplier Y. An engineer claims that microchips from Supplier Y have a lower response time than those from Supplier X. To test this claim, a random sample is taken from each supplier.

The eight microchips in the sample from Supplier X have response times, in milliseconds (ms), of:

$19.65, 19.65, 19.79, 20.75, 20.97, 21.15, 22.28, 22.37$.

Find the

(i) mean response time for the sample from Supplier X.

Find the

(ii) unbiased estimate of the population variance for the sample from Supplier X.

The seven microchips in the sample from Supplier Y have a mean response time of $19.5$ ms and an unbiased estimate of the population standard deviation ($s_{n-1}$) of $1.05$ ms.

Perform a suitable test, at the $5\%$ significance level, to test the engineer's claim that microchips from Supplier Y have a lower response time than those from Supplier X. You may assume the response times of microchips from each supplier are normally distributed with equal population variance.

(a) The formula for converting daily steps, $S$, to a fitness score, $F$, is given by $F = 0.05S + 10$.

(i) Find a formula for converting a fitness score, $F$, back to daily steps, $S$.

(ii) A user achieved a fitness score of 75. Calculate the number of daily steps they took.

(b) Over a month, the mean daily steps recorded by a group of users was 8500 steps with a standard deviation of 1200 steps.

For the same group, find

(i) the mean daily fitness score.

(ii) the standard deviation of the daily fitness scores.

(a) The scores, $s$, of 200 students on a mathematics test are recorded in the following table.

(i) Write down the mid-interval value of $60 \le s < 80$.

(ii) Calculate an estimate of the mean score of the 200 students.

(b) The data from this table is used to create a cumulative frequency graph. From this graph, the first quartile ($Q_1$) is estimated to be $60$ and the third quartile ($Q_3$) is estimated to be $96$.

Use these values to estimate the interquartile range (IQR).

(c) A student, Elara, scored $155$ on the test.

Use your answer to part (b) to estimate whether Elara's score is an outlier for this data. Justify your answer.

(d) It is believed that the scores of students on this mathematics test follow a normal distribution with mean $77.5$ and standard deviation $20$.

It is decided to perform a $\chi^2$ goodness of fit test on the data to determine whether this sample of 200 students could have plausibly been drawn from an underlying distribution $N(77.5, 20^2)$.

Write down the null and the alternative hypotheses for the test.

(e) As part of the test, the following table is created, where some categories have been combined to ensure expected frequencies are not too low.

(i) Find the value of $a$ and the value of $b$.

(ii) Hence, perform the test to a 5% significance level, clearly stating the conclusion in context.

(a) A quality control inspector at a beverage company takes a random sample of eight bottles of orange juice from a production line. The measured volumes, in ml, are:

298.5, 301.2, 299.1, 300.8, 298.9, 301.5, 299.7, 300.3

(i) Find an unbiased estimate for the mean volume of orange juice in a bottle from this production line.

(ii) Calculate a 95% confidence interval for the population mean volume. Give your answer to four significant figures.

(b) State one assumption you have made in order for your interval to be valid.

(c) The label on each bottle states: "Volume: 300 ml".

Using your answer to part (a)(ii), briefly comment on the claim on the label.