Mathematics: Applications & Interpretation SL
Statistics and regression — Topic 4
- 1.
A survey of 120 students records favourite study time (morning/evening) against year group (Year 12/Year 13). The observed frequencies give a calculated χ² test statistic of 4.83, and the critical value at the 5% significance level is 3.84. (a) State the null hypothesis for this test. (b) State whether favourite study time and year group are independent. (c) Justify your conclusion using both values. (d) State one condition required for a chi-squared test to be valid.
[4 marks] - 2.
Marking analysis: A learner attempts the following task: “A survey of 120 students records favourite study time (morning/evening) against year group (Year 12/Year 13). The observed frequencies give a calculated χ² test statistic of 4.83, and the critical value at the 5% significance level is 3.84. (a) State the null hypothesis for this test. (b) State whether favourite study time and year group are independent. (c) Justify your conclusion using both values. (d) State one condition required for a chi-squared test to be valid.” Their response addresses only this point: “States H0: favourite study time is independent of year group.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 3.
A café manager records daily temperature x (°C) and iced-drink sales y for six days: (18,20), (21,29), (24,38), (27,44), (30,55), (33,58). Using your GDC: (a) find the equation of the linear regression line of y on x. (b) State the value of Pearson's correlation coefficient r and interpret its strength. (c) Use your equation to estimate sales on a day with temperature 26°C. (d) Explain why using this equation to estimate sales at 5°C would not be appropriate.
[5 marks] - 4.
Marking analysis: A learner attempts the following task: “A café manager records daily temperature x (°C) and iced-drink sales y for six days: (18,20), (21,29), (24,38), (27,44), (30,55), (33,58). Using your GDC: (a) find the equation of the linear regression line of y on x. (b) State the value of Pearson's correlation coefficient r and interpret its strength. (c) Use your equation to estimate sales on a day with temperature 26°C. (d) Explain why using this equation to estimate sales at 5°C would not be appropriate.” Their response addresses only this point: “Uses the GDC regression feature to obtain y ≈ 2.61x − 25.9 (accept equivalent rounding).” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[5 marks] - 5.
A factory's machine produces defective components independently with probability 0.04. In a random sample of 50 components, let X be the number of defective components, assumed to follow a binomial distribution. Using your GDC: (a) state the values of n and p. (b) Calculate P(X = 2). (c) Calculate P(X ≤ 1). (d) Calculate the expected number of defective components in the sample.
[4 marks] - 6.
Marking analysis: A learner attempts the following task: “A factory's machine produces defective components independently with probability 0.04. In a random sample of 50 components, let X be the number of defective components, assumed to follow a binomial distribution. Using your GDC: (a) state the values of n and p. (b) Calculate P(X = 2). (c) Calculate P(X ≤ 1). (d) Calculate the expected number of defective components in the sample.” Their response addresses only this point: “States n = 50 and p = 0.04.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 7.
A survey of waiting times (minutes) is grouped as: 10 (frequency 4), 20 (frequency 8), 30 (frequency 10), 40 (frequency 5), 50 (frequency 3), where each value is the midpoint of its class. Using your GDC, calculate the mean and standard deviation of the waiting times.
[3 marks] - 8.
Marking analysis: A learner attempts the following task: “A survey of waiting times (minutes) is grouped as: 10 (frequency 4), 20 (frequency 8), 30 (frequency 10), 40 (frequency 5), 50 (frequency 3), where each value is the midpoint of its class. Using your GDC, calculate the mean and standard deviation of the waiting times.” Their response addresses only this point: “Enters the midpoints and frequencies into the GDC's one-variable statistics function.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] - 9.
The following 11 values are listed in ascending order: 12, 15, 18, 21, 23, 25, 27, 29, 33, 36, 40. Find the median, the lower quartile Q1, the upper quartile Q3, and the interquartile range.
[4 marks] - 10.
Marking analysis: A learner attempts the following task: “The following 11 values are listed in ascending order: 12, 15, 18, 21, 23, 25, 27, 29, 33, 36, 40. Find the median, the lower quartile Q1, the upper quartile Q3, and the interquartile range.” Their response addresses only this point: “States the median as the 6th value, 25.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 11.
A multiple-choice test has 15 questions, each with 4 options and only one correct answer. A student guesses randomly on every question. Let X be the number of correct guesses, which follows a binomial distribution. Calculate the probability that the student gets at least 5 questions correct.
[3 marks] - 12.
Marking analysis: A learner attempts the following task: “A multiple-choice test has 15 questions, each with 4 options and only one correct answer. A student guesses randomly on every question. Let X be the number of correct guesses, which follows a binomial distribution. Calculate the probability that the student gets at least 5 questions correct.” Their response addresses only this point: “States n = 15 and p = 0.25 (probability of guessing correctly on any one question).” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] - 13.
A machine fills bottles with a volume that is normally distributed with mean 60 ml. It is known that 85% of bottles contain less than 70 ml. Find the standard deviation of the filling volume, giving your answer to 3 significant figures.
[4 marks] - 14.
Marking analysis: A learner attempts the following task: “A machine fills bottles with a volume that is normally distributed with mean 60 ml. It is known that 85% of bottles contain less than 70 ml. Find the standard deviation of the filling volume, giving your answer to 3 significant figures.” Their response addresses only this point: “Uses the inverse normal function to find the z-value corresponding to a cumulative probability of 0.85, obtaining z ≈ 1.0364.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 15.
Six judges rank six gymnasts (A to F) in a competition. The ranks given by Judge 1 and Judge 2 are: Judge 1: A=1, B=2, C=3, D=4, E=5, F=6. Judge 2: A=2, B=1, C=4, D=3, E=6, F=5. Calculate Spearman's rank correlation coefficient between the two judges' rankings, and interpret its value.
[4 marks] - 16.
Marking analysis: A learner attempts the following task: “Six judges rank six gymnasts (A to F) in a competition. The ranks given by Judge 1 and Judge 2 are: Judge 1: A=1, B=2, C=3, D=4, E=5, F=6. Judge 2: A=2, B=1, C=4, D=3, E=6, F=5. Calculate Spearman's rank correlation coefficient between the two judges' rankings, and interpret its value.” Their response addresses only this point: “Finds the difference d between each pair of ranks: 1, 1, 1, 1, 1, 1 (in absolute value).” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 17.
A six-sided die is rolled 60 times, with observed frequencies for faces 1 to 6 of: 8, 12, 9, 11, 10, 10. Test, at the 5% significance level, whether the die is fair. (a) State the null hypothesis. (b) Calculate the chi-squared test statistic. (c) Given a critical value of 11.07, state your conclusion.
[5 marks] - 18.
Marking analysis: A learner attempts the following task: “A six-sided die is rolled 60 times, with observed frequencies for faces 1 to 6 of: 8, 12, 9, 11, 10, 10. Test, at the 5% significance level, whether the die is fair. (a) State the null hypothesis. (b) Calculate the chi-squared test statistic. (c) Given a critical value of 11.07, state your conclusion.” Their response addresses only this point: “States H0: the die is fair (each face has probability 1/6).” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[5 marks] - 19.
A discrete random variable X has the probability distribution: P(X=0) = 0.1, P(X=1) = 0.3, P(X=2) = 0.4, P(X=3) = 0.2. Calculate E(X) and Var(X).
[4 marks] - 20.
Marking analysis: A learner attempts the following task: “A discrete random variable X has the probability distribution: P(X=0) = 0.1, P(X=1) = 0.3, P(X=2) = 0.4, P(X=3) = 0.2. Calculate E(X) and Var(X).” Their response addresses only this point: “Confirms the probabilities sum to 1.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 21.
A school has 120 students in Grade 10, 180 in Grade 11, and 300 in Grade 12. A stratified sample of 50 students, proportional to grade size, is to be selected. Calculate the number of students that should be sampled from each grade.
[4 marks] - 22.
Marking analysis: A learner attempts the following task: “A school has 120 students in Grade 10, 180 in Grade 11, and 300 in Grade 12. A stratified sample of 50 students, proportional to grade size, is to be selected. Calculate the number of students that should be sampled from each grade.” Their response addresses only this point: “States the total population as 120 + 180 + 300 = 600.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 23.
Two classes sit the same test. Class A has a mean score of 68 with a standard deviation of 5. Class B has a mean score of 68 with a standard deviation of 14. Explain what the difference in standard deviation tells you about the two classes' results, even though their means are equal.
[2 marks] · no calculator - 24.
Marking analysis: A learner attempts the following task: “Two classes sit the same test. Class A has a mean score of 68 with a standard deviation of 5. Class B has a mean score of 68 with a standard deviation of 14. Explain what the difference in standard deviation tells you about the two classes' results, even though their means are equal.” Their response addresses only this point: “Explains that Class A's scores are more tightly clustered around the mean of 68, while Class B's scores are much more spread out.” Evaluate the response against the complete 2-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[2 marks] · no calculator - 25.
A scientist records the growth of a plant, x days after planting, and its height y cm: (2,5), (4,9), (6,13), (8,16), (10,21), (12,24). Using your GDC: (a) find the equation of the linear regression line of y on x. (b) State the value of r and interpret the strength of the correlation. (c) Use your equation to estimate the height at x = 7 days.
[4 marks] - 26.
Marking analysis: A learner attempts the following task: “A scientist records the growth of a plant, x days after planting, and its height y cm: (2,5), (4,9), (6,13), (8,16), (10,21), (12,24). Using your GDC: (a) find the equation of the linear regression line of y on x. (b) State the value of r and interpret the strength of the correlation. (c) Use your equation to estimate the height at x = 7 days.” Their response addresses only this point: “Uses the GDC's regression feature to obtain y ≈ 1.91x + 1.27 (accept equivalent rounding).” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 27.
A large box of components contains 40% defective items. A random sample of 40 components is taken, and X is the number of defective components, which follows a binomial distribution. Calculate the mean and variance of X.
[3 marks] - 28.
Marking analysis: A learner attempts the following task: “A large box of components contains 40% defective items. A random sample of 40 components is taken, and X is the number of defective components, which follows a binomial distribution. Calculate the mean and variance of X.” Their response addresses only this point: “States n = 40 and p = 0.4.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] - 29.
The lifetime of a particular brand of light bulb is normally distributed with mean 500 hours and standard deviation 20 hours. (a) Calculate the probability that a randomly selected bulb lasts between 480 and 530 hours. (b) Find the lifetime x such that only 20% of bulbs last longer than x hours.
[4 marks] - 30.
Marking analysis: A learner attempts the following task: “The lifetime of a particular brand of light bulb is normally distributed with mean 500 hours and standard deviation 20 hours. (a) Calculate the probability that a randomly selected bulb lasts between 480 and 530 hours. (b) Find the lifetime x such that only 20% of bulbs last longer than x hours.” Their response addresses only this point: “Uses the normal distribution with mean 500 and standard deviation 20.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 31.
A survey records smoking status (smoker/non-smoker) across three age groups, with observed frequencies: Age group 1: 30 smokers, 20 non-smokers. Age group 2: 25 smokers, 35 non-smokers. Age group 3: 15 smokers, 25 non-smokers. Test, at the 5% significance level with a critical value of 5.991, whether smoking status is independent of age group. (a) State the null hypothesis. (b) Calculate the chi-squared test statistic. (c) State your conclusion.
[6 marks] - 32.
Marking analysis: A learner attempts the following task: “A survey records smoking status (smoker/non-smoker) across three age groups, with observed frequencies: Age group 1: 30 smokers, 20 non-smokers. Age group 2: 25 smokers, 35 non-smokers. Age group 3: 15 smokers, 25 non-smokers. Test, at the 5% significance level with a critical value of 5.991, whether smoking status is independent of age group. (a) State the null hypothesis. (b) Calculate the chi-squared test statistic. (c) State your conclusion.” Their response addresses only this point: “States H0: smoking status is independent of age group.” Evaluate the response against the complete 6-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[6 marks]