IB Maths AI HL AI 4.4 Hypothesis testing and confidence intervals 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.

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Paper 1 Paper 2

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Question 1

Paper 1Medium6 marks

A parcel-delivery company defines its daily forecast error as

\text{actual number of parcels}-\text{forecast number of parcels}.

A random sample of 16 independent days has a mean forecast error of 18.4 parcels and a sample standard deviation of 12.0 parcels. The daily forecast errors may be modelled by a normal population, whose standard deviation is unknown.

(a)

State which distribution should be used to construct a confidence interval for the population mean forecast error, $\mu$

[1]

(b)

Calculate a 95% confidence interval for $\mu$ Give both endpoints to one decimal place.

[3]

(c)

The planning director claims that the forecasts are unbiased, so that $\mu=0$ Use your interval to comment on this claim and state the direction of any bias indicated.

[2]

Question 2

Paper 1Hard6 marks

A field team measures the nitrate concentration in water samples from a catchment. The eight samples are selected independently at random. Assume that the population of nitrate concentrations is normally distributed, with known standard deviation $1.2\text{ mg L}^{-1}$

Figure 1 shows the measured concentrations. The team will test whether the population mean concentration exceeds $5.0\text{ mg L}^{-1}$ using

H_0:\mu=5.0,\qquad H_1:\mu>5.0,

where $\mu$ is measured in $\text{mg L}^{-1}$

(a)

Use Figure 1 to determine the sample mean concentration.

[1]

(b)

Using your answer to part (a), carry out the stated test at the 5% significance level. State your conclusion in context.

[2]

(c)

Suppose that the true population mean concentration is $5.8\text{ mg L}^{-1}$ Determine the probability of a Type II error for the test in part (b). Give your answer to three significant figures.

[3]

Question 3

Paper 1Hard6 marks

During fieldwork, students randomly selected 50 plots in each of three habitat types and recorded whether a particular plant species was present.

Habitat type	Plant present	Plant absent
Woodland interior	22	28
Woodland edge	35	15
Open scrub	23	27

(a)

State appropriate null and alternative hypotheses for investigating whether plant presence is related to habitat type.

[2]

(b)

Using an appropriate statistical test and your GDC, determine the test statistic, its degrees of freedom and the p-value.

[2]

(c)

At the 5% significance level, state what the data indicate. A volunteer claims that the test proves woodland edge causes a greater probability of plant presence than open scrub. Assess this claim.

[2]

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