Statistics and Probability — Free Maths AA HL Practice Questions

Calculate the mean number of days of sick leave. [2 marks]

The value $20$ is considered a potential outlier. Evaluate whether removing this value would have a greater effect on the median or the mean, and hence determine which measure is more appropriate for summarising the original dataset. [2 marks]

Given that the mean weight is $5.0$ kg, find the value of $x$. [2 marks]

A seventh bag is added. The mean of all seven bags remains $5.0$ kg. Find the weight of the seventh bag and determine whether the mode of all seven weights differs from the mode of the original six bags. Justify your answer. [3 marks]

Using Pearson's quartile coefficient of skewness, $$S_k = \frac{Q_3 + Q_1 - 2Q_2}{Q_3 - Q_1},$$ calculate $S_k$ for each treatment and state which distribution is more skewed. [3 marks]

The combined data from both treatments are displayed in a histogram with six bins of equal width starting at 0 days. The frequencies are: Interval (days) — $[0,5)$ — $[5,10)$ — $[10,15)$ — $[15,20)$ — $[20,25)$ — $[25,30)$ Frequency — 3 — 9 — 15 — 14 — 11 — 3 Determine whether thistogram is consistent with the two individual box plots. In your answer, check the total number of patients, the range, and the position of the combined median. [3 marks]

Show that the mean height, correct to the nearest integer, is $180\,\text{cm}$. [2 marks]

The median of the data set is also $180\,\text{cm}$. Evaluate whether the mean or the mode is a more representative measure of central tendency for this data. Justify your answer with reference to the distribution of the values. [2 marks]

Calculate the probability that the first marble selected is either red or blue. [2 marks]

Given that the first marble selected is red, calculate the probability that the second marble selected is also red. [2 marks]

Find $P(R \mid B)$. [2 marks]

Determine whether $R$ and $B$ are independent events. Justify your answer. [1 mark]

Determine the probability that the person has the condition, given that the test result is positive. [3 marks]

Show that the probability that the person has the condition, given that the test result is negative, is approximately $1.06 \times 10^{-4}$. [2 marks]

A health authority proposes using this test as a mass screening tool, concluding that a positive result is strong evidence of the condition. Using your answers to parts (b) and (c), evaluate this conclusion. [2 marks]

Calculate $P(A \cap B)$. [2 marks]

Hence determine whether events $A$ and $B$ are independent. Justify your answer. [2 marks]

Find the $p$-value for the test, showing your working clearly. [3 marks]

Show that the $t$-statistic for this sample is $t = -1.634$, correct to three decimal places. [3 marks]

State the critical value for this test and hence determine, with justification, whether the researcher has sufficient evidence to support the company's claim at the $5\%$ significance level. [2 marks]

Construct a 95% confidence interval for $\mu$. [2 marks]

The company claims the drug reduces systolic blood pressure by an average of at least 10 mmHg. Using a one-tailed $t$-test at the 5% significance level, determine whether the sample evidence supports this claim. State your hypotheses, calculate the test statistic, and state your conclusion in context. [4 marks]

Calculate the $p$-value for this test. [3 marks]

The researcher uses a significance level of $5\%$. State the conclusion of the test. Justify whether the result is practically significant, given that a difference of less than $0.5$ hours in mean sleep time is considered negligible in clinical terms. [3 marks]

Using your value of $x$ from part (a), calculate the interquartile range of the heights. [2 marks]

After the squad is selected, the player of height $194\,\text{cm}$ is injured and replaced by a new player of height $210\,\text{cm}$. Determine whether the mean and the median each increase, decrease, or stay the same. Justify both answers. [3 marks]

For this value of $y$, calculate the mean overtime hours, giving your answer correct to $3$ significant figures. [2 marks]

A new employee joins the company. Their overtime hours, $k$, are added to the dataset (now $11$ values). Determine the range of integer values of $k$ for which the mean of the new dataset exceeds the median of the new dataset. [4 marks]

Determine the median height of the sample. [2 marks]

The biologist adds a 13th plant to the sample. Its height is $h$ cm, where $25 < h < 27$. Determine the new mean and new median, giving your answers in terms of $h$ where appropriate, and justify whether the median increases, decreases, or stays the same compared to part (b). [2 marks]

Calculate the variance of the heights. Give your answer correct to 3 significant figures. [3 marks]

A 13th sunflower plant is added to the data set. Its height is $28\,\text{cm}$. Calculate the new mean and standard deviation of all 13 heights. Give your answers correct to 3 significant figures. [3 marks]

Determine, with a reason, whether events $A$ and $B$ are independent. [2 marks]

A component is selected at random and found to have defect $A$. Calculate the probability that it also has defect $B$. [1 mark]

A quality-control engineer claims: *"A component with defect $A$ is more than twice as likely to have defect $B$ as a randomly selected component."* Determine whether this claim is correct, justifying your answer with a comparison of appropriate probabilities. [1 mark]

Calculate the probability that the student plays exactly one of the two sports. [2 marks]

Given that the student does not play basketball, find the probability that they play football. Hence determine, with justification, whether the events $F$ and $B'$ are independent. [2 marks]

Given that a randomly selected component is defective, determine the probability that it was produced by machine A. [2 marks]

The factory manager claims that, because machine A produces the majority of components, it is responsible for the majority of defective components. Evaluate this claim. [1 mark]

Determine $P(D \mid T)$, the probability that a patient who tests positive actually has the disease. Give your answer correct to three significant figures. [2 marks]

A health authority considers the test effective only if a patient who tests positive has at least a $50\%$ probability of actually having the disease. Determine the minimum true positive rate required, assuming all other probabilities remain unchanged, for the test to meet this criterion. [4 marks]

Calculate the value of the test statistic. [2 marks]

The $p$-value for the test is $0.0465$ (to 3 significant figures). A statistician argues that, because the $p$-value is only just below $0.05$, the result provides weak evidence against $H_0$ and the company's claim should not yet be acted upon. Evaluate this argument. [3 marks]

Calculate the value of the $t$-test statistic. [2 marks]

State the critical values for this test and hence determine, with justification, whether the manager should reject the null hypothesis. [2 marks]

State the degrees of freedom and determine the critical value for a one-tailed test at the $5\%$ significance level. [1 mark]

State whether the researcher has sufficient evidence to reject the company's claim. Justify your answer. [2 marks]

Determine the degrees of freedom and the critical values for this two-tailed test. [2 marks]

State the conclusion of the test. Justify whether the researcher's result would change if the significance level were increased to $5\%$. [2 marks]

A researcher claims that neither zone's rainfall data is symmetrically distributed. For each zone, calculate $$d = \frac{(Q_3 - \text{Median}) - (\text{Median} - Q_1)}{Q_3 - Q_1}$$ and the ratio of upper whisker length to lower whisker length. Hence evaluate the researcher's claim, stating the direction of skew for each zone. [5 marks]

A histogram of Zone A's rainfall is constructed using 8 equal-width bins spanning $300\,\text{mm}$ to $800\,\text{mm}$. The total frequency is 100. Assuming rainfall is uniformly distributed within each inter-quartile interval, show that the frequency density of the bin $[487.5, 550)$ is approximately $0.319\,\text{mm}^{-1}$, and hence prove that this frequency density is less than $0.4\,\text{mm}^{-1}$. [3 marks]

For each treatment, estimate $Q_1$, $Q_3$, and the interquartile range (IQR), assuming data are uniformly distributed within each bin. Hence determine which treatment has the larger IQR. [5 marks]

The researcher claims that the median recovery time for Treatment Y is greater than that for Treatment X. Determine, showing all working, whether this claim is correct. [3 marks]

(i) Find $P(C)$. [3 marks]
(ii) Prove that events $C$ and $D$ are not independent. [3 marks]

Show that the overall probability that a randomly selected component is defective is $0.019$. [2 marks]

A component is selected at random and found to be defective. Using Bayes' theorem, calculate $P(A \mid D)$ and $P(C \mid D)$, where $D$ is the event that the component is defective. [4 marks]

The quality-control manager claims that machine C poses a greater risk to overall product quality than machine A, both in terms of posterior likelihood of being the source of a defect and in terms of defect rate per unit produced. Evaluate this claim. [2 marks]

(i) The researcher uses the critical value $r_{\text{crit}} = 0.707$ for a two-tailed test at the $5\%$ significance level with $n = 8$. State the hypotheses and determine whether the correlation is statistically significant at this level. [2]
(ii) An alternative approach uses the test statistic $$t = r\sqrt{\frac{n-2}{1-r^2}}$$ which follows a $t$-distribution with $n - 2$ degrees of freedom under $H_0$. Calculate $t$ and hence verify the conclusion from (b)(i). Evaluate whether the researcher's claim of statistical significance is reliable, with reference to the assumptions required for Pearson's correlation coefficient to be valid. [4 marks]

Given that $R^2 = \dfrac{S_{xy}^2}{S_{xx}\,S_{yy}}$, show that $R^2 = 0.64$. [2 marks]

State what the value $R^2 = 0.64$ indicates about the linear model. [1 mark]

A researcher proposes using the regression model to predict the reaction rate at $x = 50\,°\text{C}$. Evaluate the validity of this prediction, with reference to the assumptions of linear regression and the consequences of extrapolation. [4 marks]