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 study examines whether the type of packaging (plastic, glass, or metal) affects the shelf life (short, medium, or long) of a product. Data from 180 products are recorded:
| Short | Medium | Long | |
|---|---|---|---|
| Plastic | 20 | 25 | 15 |
| Glass | 10 | 30 | 20 |
| Metal | 15 | 25 | 20 |
A test is conducted at the significance level to test for independence.
Calculate the expected frequency for products in glass packaging with a medium shelf life.
Find the degrees of freedom and the critical value for this test.
Given the p -value is 0.008 , state the conclusion of the test and justify your answer.
Explain what a Type I error means in this context.
A goodness of fit test is performed at the significance level to test if the data follow a Poisson distribution with mean 2.
| Errors | 0 | 1 | 2 | 3 | 4 or more |
|---|---|---|---|---|---|
| Frequency | 40 | 80 | 90 | 60 | 30 |
Calculate the expected frequency for 2 errors.
Find the degrees of freedom for this test.
Given the test statistic is 7.85 , determine whether the data can be modeled by a Poisson distribution with mean 2. Justify your answer.
A survey of 150 students at a school records their preferred extracurricular activity (sports, arts, or music) and their year group (Year 10 or Year 11). The results are shown in the table below:
| Sports | Arts | Music | |
|---|---|---|---|
| Year 10 | 30 | 20 | 15 |
| Year 11 | 25 | 35 | 25 |
A test is conducted at the significance level to determine if the preferred activity is independent of the year group.
State the null hypothesis, , for this test.
Calculate the expected frequency for Year 10 students choosing sports.
Find the degrees of freedom for this test.
Given the test statistic is 6.25 , determine the conclusion of the test. Justify your answer using the critical value or p-value.
A teacher records the time (in minutes) taken by two classes, Class X and Class Y , to complete a coding task. The data are normally distributed. The sample statistics are:
A t-test is conducted at the 1% significance level to determine if Class X takes longer on average than Class Y.
State the null and alternative hypotheses for this test.
Explain what a Type II error means in this context.
Given the t-test statistic is 2.10 with 53 degrees of freedom, find the p-value and state the conclusion of the test.