IB Maths AI SL Topic 4: Statistics and probability questions

State the population and the sample.

Calculate the interquartile range and determine whether the 18-hour repair time is an outlier.

The maintenance log confirms that this repair took 18 hours because a lane closure delayed the crew. State whether the value should be removed before describing the sample, giving a reason.

Write down the median commute time for Route A.

Calculate the interquartile range of the commute times for Route B. Give your answer in minutes.

State which route’s commute times may be normally distributed. Give one feature of its box-and-whisker diagram to support your answer.

Calculate the interquartile range for each route.

State which route has the higher median delivery time.

State which route’s delivery times may be normally distributed. Give a reason based on the symmetry of its box and whiskers.

Use a GDC to calculate Pearson’s product-moment correlation coefficient, $r$ to three significant figures. Describe the direction and strength of the linear correlation.

Use a GDC to find the equation of the regression line of $V$ on $t$ giving its coefficients to four significant figures.

Use your regression line to estimate the volume of water after 9 minutes. Give your answer in litres to three significant figures.

Interpret the gradient and the vertical intercept of the regression line in this context.

Use a graphic display calculator to find Spearman’s rank correlation coefficient, $r_s$ for these data. Give your answer to three significant figures.

Interpret your value of $r_s$ in the context of the fieldwork.

The scatter plot suggests an overall increasing, curved relationship between distance from the road and the number of native plant species. State which coefficient, Pearson’s product moment correlation coefficient or Spearman’s rank correlation coefficient, is more appropriate for describing this relationship. Give a reason.

If one additional observation is an extreme value, explain how Pearson’s product moment correlation coefficient and Spearman’s rank correlation coefficient are likely to differ in their sensitivity to it.

Use a graphic display calculator to find Spearman’s rank correlation coefficient, $r_s$ for these data. Give your answer to three significant figures.

State the direction of the association indicated by $r_s$

If the researchers want to assess a linear relationship, rather than a monotonic relationship in general, name the more appropriate coefficient: Pearson’s product-moment correlation coefficient or Spearman’s rank correlation coefficient.

Plot F appears to be an outlier. State which of these two coefficients is generally less sensitive to an outlier.

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.

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.

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

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.

Sketch the normal curve for $X$ Label the horizontal axis with $\mu-\sigma$ $\mu$ $\mu+\sigma$ and their corresponding masses.

Use your GDC to calculate the probability that a packet has a mass between $496$ g and $504$ g. Give your answer to four significant figures.

Use your GDC to find the mass $x$ such that $90\%$ of packets have a mass less than $x$ Give your answer to the nearest $0.1$ g.

Write down $P(X=1)$

Show that $E(X)=0$

Hence state whether the game is fair.

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

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.

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.

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.

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.

For the comparison of mean dissolved oxygen concentrations:

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.

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.

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

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.

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.