Practice SL 4.11—Expected, observed, hypotheses, chi squared, gof, t-test with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.
A researcher investigates whether the mean reaction time (in milliseconds) of two groups of participants, Group A (using a new training method) and Group B (using a standard method), differs. The data are normally distributed with equal variances. The sample statistics are:
A two-sample t -test is conducted at the significance level.
State the null and alternative hypotheses for this test.
Find the probability that a participant from Group A has a reaction time less than 300 ms .
Given the p -value for the t -test is 0.073 , state the conclusion of the test and justify your answer.
A sports scientist compares sprint times (in seconds) of athletes using two training programs, A and B. The data are normally distributed with equal variances. Sample statistics:
A t-test is conducted at the significance level. The scientist also examines the combined data for normality.
State the null and alternative hypotheses, and explain why a t -test is appropriate.
Calculate the probability that an athlete from Program A runs faster than 10.5 seconds.
Compute the t-test statistic and find the p-value.
Conclude the t-test, and discuss the implications for training program selection.
Combine the samples and test if the combined times are normally distributed using a goodness of fit test at the significance level. Intervals: , . Observed frequencies: . Estimate the mean and standard deviation, and calculate the expected frequency for .
Given the statistic is 10.25 , conclude the goodness of fit test.
A researcher tests whether a new fertilizer affects the number of flowers produced by a plant species, assumed to follow a Poisson distribution. Data from 200 plants are summarized:
| Flowers | 0 | 1 | 2 | 3 | 4 or more |
|---|---|---|---|---|---|
| Frequency | 25 | 50 | 60 | 45 | 20 |
A goodness of fit test is performed at the significance level, with the Poisson mean estimated from the data.
Estimate the Poisson mean using the sample data.
Calculate the expected frequency for 1 flower, using the estimated mean.
Given the test statistic is 5.12 with 3 degrees of freedom, state the conclusion of the test. Justify your answer.
A researcher collects data on the time (in minutes) it takes for two groups of students, Group X and Group Y, to complete a math puzzle. The sample means and standard deviations are:
The researcher conducts a two-sample t-test at the significance level to determine if there is a difference in the mean completion times between the two groups.
State the null and alternative hypotheses for this test.
Explain what is meant by a Type I error in this context.
Given that the p -value is 0.015 , state the conclusion of the test and justify your answer.
A quality control team tests whether the number of defects in a manufacturing process follows a Poisson distribution. Over 500 units, defects are recorded:
| Defects | 0 | 1 | 2 | 3 | 4 | 5 or more |
|---|---|---|---|---|---|---|
| Frequency | 120 | 180 | 100 | 60 | 30 | 10 |
A goodness of fit test is conducted at the significance level, with the Poisson mean estimated from the data. The team also analyzes defect probabilities.
(a) Estimate the Poisson mean using the sample data.
(b) Calculate the expected frequency for 2 defects using the estimated mean.
(c) Find the degrees of freedom for the test.
(d) Compute the test statistic, showing all steps.
(e) Conclude the test using the critical value or p -value.
(f) Calculate the probability that a randomly selected unit has at most 1 defect, using the estimated mean.
(g) If two units are selected with replacement, find the probability that both have exactly 2 defects.
The heights of seedlings in two different greenhouses, A and B, are assumed to follow normal distributions. Greenhouse A has a mean height of 12 cm with a standard deviation of 1.5 cm , and Greenhouse B has a mean height of 11 cm with a standard deviation of 1.2 cm . A gardener performs a t-test to determine if the mean height of seedlings in Greenhouse is greater than that in Greenhouse at the significance level.
State the null and alternative hypotheses for this test.
Find the probability that a seedling from Greenhouse A has a height greater than 13 cm.
Given that the p -value for the t -test is 0.085 , state the conclusion of the test and justify your answer.
A researcher is studying the distribution of blood types in a population. According to national statistics, the expected distribution of blood types is: Type O (44%), Type A (42%), Type B (10%), and Type AB (4%). In a sample of 200 people from a specific region, the following frequencies were observed: Type O: 75, Type A: 92, Type B: 24, Type AB: 9.
State the null and alternative hypotheses for this test.
Calculate the expected frequencies for each blood type.
Calculate the chi-squared test statistic. Show your working.
Using a 5% significance level, state the critical value for this test and make a conclusion.
A newspaper vendor in Singapore is trying to predict how many copies of The Straits Times they will sell. The vendor forms a model to predict the number of copies sold each weekday. According to this model, they expect the same number of copies will be sold each day. To test the model, they record the number of copies sold each weekday during a particular week. This data is shown in the table.
| Day | Monday | Tuesday | Wednesday | Thursday | Friday |
|---|---|---|---|---|---|
| Number of copies sold | 74 | 97 | 91 | 86 | 112 |
A goodness of fit test at the 5% significance level is used on this data to determine whether the vendor's model is suitable. The critical value for the test is 9.49 and the hypotheses are H₀: The data satisfies the model. H₁: The data does not satisfy the model.
Find an estimate for how many copies the vendor expects to sell each day.
Write down the degrees of freedom for this test.
Write down the conclusion to the test. Give a reason for your answer.
At Springfield University, the weights, in kg, of 10 chinchilla rabbits and 10 sable rabbits were recorded. The aim was to find out whether chinchilla rabbits are generally heavier than sable rabbits. The results obtained are summarized in the following table.
| Weight of chinchilla rabbits, kg | 4.9 | 4.2 | 4.1 | 4.4 | 4.3 | 4.6 | 4.0 | 4.7 | 4.5 | 4.4 |
|---|---|---|---|---|---|---|---|---|---|---|
| Weight of sable rabbits, kg | 4.2 | 4.1 | 4.1 | 4.2 | 4.5 | 4.4 | 4.5 | 3.9 | 4.2 | 4.0 |
A -test is to be performed at the 5% significance level.
Write down the null and alternative hypotheses.
Find the -value for this test.
Write down the conclusion to the test. Give a reason for your answer.
A newspaper vendor in Singapore is trying to predict how many copies of The Straits Times they will sell. The vendor forms a model to predict the number of copies sold each weekday. According to this model, they expect the same number of copies will be sold each day. To test the model, they record the number of copies sold each weekday during a particular week. This data is shown in the table.
| Day | Monday | Tuesday | Wednesday | Thursday | Friday |
|---|---|---|---|---|---|
| Number of copies sold | 74 | 97 | 91 | 86 | 112 |
A goodness of fit test at the 5% significance level is used on this data to determine whether the vendor's model is suitable. The critical value for the test is 9.49.
Find an estimate for how many copies the vendor expects to sell each day.
State the null and alternative hypotheses for this test.
Write down the degrees of freedom for this test.
Write down the conclusion to the test. Give a reason for your answer.