Statistics Exam Questions: Math and Vocabulary Datasets

Mathematics Exam Questions: Focusing on the Subtopic of Statistics

Mathematics examinations often feature a section dedicated to statistics. These questions aim to assess students' understanding of various statistical concepts and their ability to apply them to solve problems. Here, we focus on a selection of statistics exam questions from different sources. Let's dive into some examples!

Question 1

Consider the dataset of children's scores on two exams: math and vocabulary. Each child takes one exam on each subject, with larger scores being better for both subjects. The data is normally distributed.

(a) Find the mean math score and the median vocabulary score. Answer: The mean math score is 15, and the median vocabulary score is 18.

(b) For this dataset, what value of the standard deviation (s) will contain approximately 68% of the data? Hint: Remember that the empirical rule states that about 68% of the data falls within one standard deviation from the mean.

(c) If we have another dataset where the distribution is strongly skewed to the right, which value of s would you expect to contain 80% or more of the data?

Question 2

Suppose Al and Bev take two different tests each (one math test and one vocabulary test). Al's total score on all four tests is 30, while Bev's total score is 90. Calculate Carol's total and mean if her scores are equal parts from each subject taken by either Al or Bev.

(a) What is Carol's total score? Answer: Carol's total score is 120.

(b) What is Carol's mean score? Hint: To find the mean, divide the total by the number of subjects.

Question 3

Find the range for the following dataset of grades: {63.0, 64.3}. Also calculate the median using the data from a single test taken by any student.

(a) What is the range of grades? Answer: The range is 1.3.

(b) Calculate the median using the provided data from one test taken by any student. Hint: Since there are only two values, the median is simply their average.

Question 4

Given two sets of sorted numbers representing scores from two populations, calculate Gosset's 90% confidence interval for the mean of the first population.

(a) Calculate the minimum and maximum values for the first set of numbers. Hint: The range represents the difference between the highest and lowest values.

(b) Calculate the mean and standard deviation for both sets of numbers. Hint: To find the mean, add all the values together and divide by the number of scores. For the standard deviation, use the formula s = sqrt(Σ (xi − x̄)^2 / n - 1), where xi is each individual score, x̄ is the mean, s is the standard deviation, Σ is the sum of the squared differences between each score and the mean, and n is the number of scores.

(c) Calculate Gosset's 90% confidence interval for the mean of the first population. Hint: Use the formula μ = x̄ ± t(α/2)s/√n, where μ is the population mean, x̄ is the sample mean, t(α/2) is the critical value from a t-distribution with ν degrees of freedom, s is the standard deviation of the sample, and n is the number of scores in the sample. For this problem, use α = 0.10 and ν = 30 (the recommended degrees of freedom for a normal distribution).