Hector guesses on each question of a 35 question multiple choice test. Each question has 5 possible responses. What are the mean and standard deviation of the number of questions that Hector will guess correctly?
Mean = S.D. = [$\square$]
Round to the nearest hundredth.
Each guess is independent of the other guesses.
Answer (2)
Recognize the problem as a binomial distribution with n = 35 and p = 5 1 = 0.2 .
Calculate the mean using the formula: Mean = n × p = 35 × 0.2 = 7 .
Calculate the standard deviation using the formula: Standard Deviation = n × p × ( 1 − p ) = 35 × 0.2 × 0.8 = 5.6 ≈ 2.37 .
State the mean and standard deviation: Mean = 7 and Standard Deviation ≈ 2.37 .
Explanation
Analyze the problem Let's analyze the problem. Hector is guessing on a multiple-choice test, and we want to find the mean and standard deviation of the number of questions he'll get right. This is a binomial distribution because each question is an independent trial with a fixed probability of success (guessing correctly).
Identify given values We have n = 35 questions, and the probability of guessing correctly on each question is p = 5 1 = 0.2 , since there are 5 possible answers for each question.
Calculate the mean The mean (expected value) for a binomial distribution is given by the formula: Mean = n × p Substituting the values, we get: Mean = 35 × 0.2 = 7
Calculate the variance The variance for a binomial distribution is given by the formula: Variance = n t im es p × ( 1 − p ) Substituting the values, we get: Variance = 35 × 0.2 × ( 1 − 0.2 ) = 35 × 0.2 × 0.8 = 5.6
Calculate the standard deviation The standard deviation is the square root of the variance: Standard Deviation = Variance Standard Deviation = 5.6 ≈ 2.36643 Rounding to the nearest hundredth, we get 2.37.
State the final answer Therefore, the mean number of questions Hector is expected to guess correctly is 7, and the standard deviation is approximately 2.37.
Examples
Consider a quality control process where a certain percentage of products are expected to be defective. If you inspect a batch of products, you can use the binomial distribution to calculate the mean and standard deviation of the number of defective items you might find. This helps in understanding the expected range of defective products and making informed decisions about the quality of the production process.
Hector is expected to guess correctly on 7 out of 35 questions, with a standard deviation of approximately 2.37. These values are calculated using the binomial distribution formulas for mean and standard deviation. This approach allows us to understand Hector's performance on this test statistically.
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