IB Mathematics Applications & Interpretation SL 4.11—expected, Observed, Hypotheses, Chi Squared, Gof, T-test Questionbank

(d) State your conclusion for the t-test and comment on the choice between programs A and B.

(f) 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.

(a) 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.

Formulate null and alternative hypotheses and conduct a goodness-of-fit test at the 5% significance level to evaluate if the distribution follows $B(4,0.65)$.

(c) In a separate sample of class bookings, the membership type and whether the member attended the class were recorded.

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.

(a) State the null and alternative hypotheses for this test.

(b) Calculate the expected frequencies for each blood type.

(c) Calculate the chi-squared goodness-of-fit (GOF) test statistic. Show your working.

(d) Using a 5% significance level, state the critical value for this test and make a conclusion.

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.

(b) Calculate the expected frequencies for each snack type.

A transit authority is evaluating use of self-service ticket kiosks.

(a) At one station, 120 commuters each needed to buy a single ticket on 4 separate mornings. For each commuter, the number of mornings on which the self-service kiosk was used was recorded.

Formulate null and alternative hypotheses and conduct a goodness-of-fit test at the 5% significance level to evaluate if the distribution follows $B(4,0.6)$.

(c) In a separate sample of kiosk transactions, the payment method and the outcome were recorded.

Perform a chi-squared test for independence at the 5% significance level to determine if there is a significant association between payment method and transaction outcome.

(a) State the null and alternative hypotheses for this test.

(c) Calculate the chi-squared goodness-of-fit (GOF) test statistic. Show your working.

(d) Using a 5% significance level, state the critical value for this test and make a conclusion.

(e) 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 $(11,12]$. Intervals: $(-\infty, 10],(10,11]$, $(11,12],(12,13],(13, \infty)$. Observed frequencies: $5,15,25,15,5$.

(b) Without performing additional calculations, briefly describe the procedure to test, at the 5% level of significance, if the data is consistent with a $B(4,p)$ distribution, where the probability $p$ is unknown.

(b) Without performing additional calculations, briefly describe the procedure to test, at the 5% level of significance, if the data is consistent with a $B(4,p)$ distribution, where the probability $p$ is unknown.