4.12 Data collection and reliability questions for IB Maths AI

Explain what is wrong with Question 1 and rewrite it so that it is unbiased.

Explain what is wrong with Question 2 and suggest how it could be improved.

Explain what is wrong with the answer choices in Question 3 and rewrite them.

Find the mean number of goals scored by the home team per match.

Find the unbiased estimate of the population variance and hence comment on whether a Poisson distribution is likely to be a suitable model.

Bilal decides to test whether the number of goals scored by the home team can be modelled by a Poisson distribution with mean equal to the value found in part (a). The expected frequencies are shown in the table, correct to two decimal places, where the final column is for 5 or more goals.

Find the value of $p$ and the value of $q$.

Explain why Bilal combines the last two columns before carrying out a ${\chi }^2$ goodness of fit test, and hence write down the number of degrees of freedom for his test.

Perform the test at the 5% significance level. State your hypotheses and conclusion clearly.

Using the Poisson model, find the probability that the home team scores at least 2 goals in a randomly chosen match.

Explain the difference between the reliability and the validity of the test.

Describe how Dr Okonkwo could use the test–retest method to assess the reliability of her test.

Suggest how she could assess the criterion-related validity of the test.

Find
(i) the mean number of dolphins per interval;
(ii) an unbiased estimate of the population variance.

Explain how your answers to part (a) support a Poisson model for the number of dolphins in a 10-minute interval.

Aroha models the number of dolphins in an interval by $X\sim \mathrm{P}\mathrm{o}(1.84)$. The expected frequencies are shown in the table, where $X\ge 5$ is used for the last class.

Find the value of $j$ and the value of $k$.

Explain why the last two classes should be combined before a ${\chi }^2$ goodness of fit test is performed.

Show that the number of degrees of freedom for the test is 3.

Perform the test at the 5 % significance level, stating your hypotheses and your conclusion.

Use the model to find the probability that at least 10 dolphins pass the lookout point in one hour.

Write down the name of this method of assessing reliability.

Find Pearson's product-moment correlation coefficient, $r$, between the first and second scores.

Comment on the reliability of the test.

The human resources manager says: "These results prove that the test is a valid way of choosing the best employees." Comment on this statement.

Calculate the expected frequency of low-dose patients with severe side effects.

Explain why the "Severe" and "Moderate" columns should be combined before carrying out the test.

Write down the new table of observed frequencies, and the number of degrees of freedom for the test.

Perform the test, stating your hypotheses and your conclusion in context.

Dr Lindqvist notices that the low-dose group had more severe or moderate side effects than expected. Suggest a reason, related to how the trial was designed, why this result should be treated with caution.