IB Maths AI SL Topic 4: Statistics and probability questions

Exam-style questions in the shape of Paper 1 and Paper 2, with full markschemes. Read the markscheme, or type, photograph or draw your answer and get it marked.

Paper

Paper 1 Paper 2

Difficulty

Easy Medium Hard

Question 1

Paper 2Hard16 marks

An environmental science team records water clarity and aquatic plant cover at eight ponds in one catchment. The team wants to investigate the association between these variables.

Pond	Water clarity (cm)	Aquatic plant cover (%)
A	10	12
B	20	18
C	30	25
D	30	31
E	40	39
F	50	45
G	60	52
H	200	10

Field notes state that pond H was recently dredged. Measurements at H were repeated and agreed.

(a)

Show that ponds C and D are each assigned a clarity rank of 3.5 when the clarity values are ranked from smallest to largest.

[2]

(b)

Using all eight ponds and your GDC, determine Pearson’s product moment correlation coefficient, $r$ and Spearman’s rank correlation coefficient, $r_s$ Give both coefficients to three significant figures.

[4]

(c)

Calculate $r$ and $r_s$ again, using the seven ponds other than H. Give both coefficients to three significant figures.

[4]

(d)

A researcher claims that the full-data Pearson coefficient proves there is no association between clarity and plant cover. The researcher also claims that H should be removed as an outlier and that the Pearson coefficient for the remaining ponds should be reported as the relationship.

Discuss these claims and recommend how the results should be reported. Refer to the coefficients and the field notes.

[6]

Question 2

Paper 1Hard8 marks

At a cereal plant, the mass $X$ in grams, of a randomly selected bag is modelled by a normal distribution with mean $502$ g and standard deviation $4.8$ g. A bag is accepted if its mass is at least $500$ g.

The machine can be adjusted by adding the same amount $d$ grams to the mass of every bag. This adjustment does not change the standard deviation.

(a)

Write down the interval that contains approximately $68\%$ of the bag masses at the current setting.

[1]

(b)

Use your GDC to find the probability that a randomly selected bag is accepted at the current setting. Give your answer to four significant figures.

[2]

(c)

The manager requires at least $97.5\%$ of the bags to be accepted. Using your answer to (b), state whether the current setting meets this requirement. Determine the smallest value of $d$ that meets the requirement. Give your answer to three significant figures.

[5]

Question 3

Paper 2Hard14 marks

An environmental field team compared aquatic indicator-plant abundance in one restored wetland and one unrestored wetland. At each wetland, 60 quadrats were selected at random. The number of indicator-plant species in each quadrat was classified as low, medium or high.

Wetland	Low	Medium	High
Restored	12	30	18
Unrestored	24	18	18

(a) Calculate the expected frequencies for all six cells if wetland type and abundance class are independent. State whether the expected-frequency condition for a chi-squared test is satisfied. [2]

(b) For a chi-squared test of independence at the 5% significance level, state H₀ and H₁, give the degrees of freedom, and use your GDC to find the chi-squared statistic and p-value. [5]

Dissolved oxygen was measured in a randomly selected subset of 24 quadrats at each wetland. The summary statistics were:

Wetland	Sample size	Mean dissolved oxygen (mg L⁻¹)	Sample standard deviation (mg L⁻¹)
Restored	24	7.1	1.0
Unrestored	24	6.5	1.2

Before collecting the data, the team predicted that the mean dissolved oxygen concentration would be higher at the restored wetland. Assume that the two samples are unpaired, the underlying populations are normal, and their unknown variances are equal.

(i) write down H₀ and H₁ for the team’s prediction. [2]

(ii) Use a pooled two-sample t-test on your GDC to find the one-tailed p-value for this prediction and the two-tailed p-value. [2]

(d) At the 5% significance level, assess an officer’s claim that the fieldwork proves restoration caused an improvement in wetland quality. Use the results and the fieldwork design in your answer. [3]

(a)

Calculate the expected frequencies for all six cells if wetland type and abundance class are independent. State whether the expected-frequency condition for a chi-squared test is satisfied.

[2]

(b)

For a chi-squared test of independence at the 5% significance level, state H₀ and H₁, give the degrees of freedom, and use your GDC to find the chi-squared statistic and p-value.

[5]

(c)

For the comparison of mean dissolved oxygen concentrations:

(i)

Write down H₀ and H₁ for the team’s prediction.

[2]

(ii)

Use a pooled two-sample t-test on your GDC to find the one-tailed p-value for this prediction and the two-tailed p-value.

[2]

(d)

At the 5% significance level, assess an officer’s claim that the fieldwork proves restoration caused an improvement in wetland quality. Use the results and the fieldwork design in your answer.

[3]

Question 4

Paper 2Hard14 marks

A field ecologist investigates whether bee visits differ between a mown and an unmown meadow. There is one meadow of each type. For the first investigation, 30 distinct bee visits were randomly selected from each meadow and classified by duration.

Meadow	Less than 10 s	10 s to less than 20 s	20 s or more	Total
Mown	8	12	10	30
Unmown	14	10	6	30
Total	22	22	16	60

For a separate investigation, a random sample of 12 distinct bee visits was taken from each meadow. The sample means and sample standard deviations of visit duration were as follows.

Meadow	Sample size	Mean duration (s)	Sample standard deviation (s)
Mown	12	21.4	4.8
Unmown	12	16.1	5.2

Assume the two populations of visit durations are normally distributed, have equal variances, and were sampled independently. Use a 5% significance level for each test.

(a)

State the null and alternative hypotheses for a chi-squared test of independence between meadow type and visit-duration category. Show that the expected frequencies for the mown meadow are 11, 11 and 8.

[4]

(b)

Using your GDC, find the degrees of freedom, chi-squared statistic and p-value for the test. State your conclusion.

[4]

(c)

The ecologist predicts that the population mean visit duration is greater in the mown meadow. State suitable hypotheses and use your GDC to carry out an appropriate test. Give the p-value and a conclusion.

[4]

(d)

The ecologist concludes that mowing causes longer bee visits and changes the distribution of visit-duration categories. Assess this conclusion in light of the results and the fieldwork design.

[2]

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