Practice IB Mathematics Applications & Interpretation (AI) Topic SL 4.11—expected, Observed, Hypotheses, Chi Squared, Gof, T-test with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for SL 4.11—expected, Observed, Hypotheses, Chi Squared, Gof, T-test and mirrors Paper 1, 2, 3 style where relevant.
Get instant solutions, detailed explanations, and build confidence with questions aligned to IB examiner expectations.
A sports scientist compares sprint times (in seconds) of athletes using two training programs, A and B. Assume the data are approximately 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 seconds.
Compute the t-test statistic and find the p-value.
State your conclusion for the t-test and comment on the choice between programs A and B.
Combine the samples and assess whether the combined times are normally distributed using a normal probability plot. Estimate the mean and standard deviation (in seconds), and calculate the expected frequency for . Intervals: , . Observed frequencies: .
Given the normal probability plot suggests the combined data are not normally distributed, state one implication this may have for the validity of the t-test conclusion.
A community fitness centre is evaluating use of its online class-booking system.
At one branch, 160 members each reserved a place in a class on 4 separate weeks. For each member, the number of weeks on which the online booking system was used was recorded.
| Number of weeks using the system | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Frequency of members | 0 | 32 | 38 | 56 | 34 |
Formulate null and alternative hypotheses and conduct a goodness-of-fit test at the 5% significance level to evaluate if the distribution follows .
Without performing additional calculations, briefly describe the procedure to test, at the 5% level of significance, if the data is consistent with a distribution, where the probability is unknown.
In a separate sample of class bookings, the membership type and whether the member attended the class were recorded.
| Membership type | Attended | Missed |
|---|---|---|
| Standard | 60 | 20 |
| Student | 45 | 25 |
| Premium | 35 | 15 |
Perform a chi-squared test for independence at the 5% significance level to determine if there is a significant association between membership type and attendance status.
A researcher is studying the distribution of blood types in a population and wants to carry out a chi-squared goodness-of-fit (GOF) test. 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 goodness-of-fit (GOF) test statistic. Show your working.
Using a 5% significance level, state the critical value for this test and make a conclusion.
Trays at a warehouse each contain five ceramic mugs. Each mug is either intact or cracked. It is hypothesized that the number of cracked mugs in a tray follows a binomial distribution. A sample of 120 trays was inspected with the following results.
| Number of cracked mugs per tray | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| Frequency | 22 | 43 | 34 | 16 | 4 | 1 |
Find the arithmetic mean number of cracked mugs per tray.
Thus, determine an estimate for , the success probability of choosing a cracked mug.
Using a goodness of fit test, evaluate at the 5% significance level if the binomial model is an appropriate fit for this dataset.
A cinema manager is studying the distribution of snack purchases and wants to carry out a chi-squared goodness-of-fit (GOF) test. According to company-wide sales data, the expected distribution of snack purchases is: popcorn (36%), nachos (28%), chocolate (20%), and candy (16%). In a sample of 250 customers, the following frequencies were observed: popcorn: 72, nachos: 63, chocolate: 60, candy: 55.
State the null and alternative hypotheses for this test.
Calculate the expected frequencies for each snack type.
Calculate the chi-squared goodness-of-fit (GOF) test statistic. Show your working.
Using a 5% significance level, state the critical value for this test and make a conclusion.