I don't mind you showing me the caluclator key strokes to do this
or show me the mathematical way whice ever thanks
Here is a simple probability model for multiple-choice tests. Suppose that a student has probability p of correclty ansering a question chosen at random form a univers of possible questions. (A strong student has a higher p than a weak student.) The correctness of an anser to any specific question doesn't depend on other quetions. a test contains n questions. Then the proportion of correct ansers that a student gives in a sample proportion p hat form an SRS of size n dranw from a population with population proportion p.
(a) Julie is a good student for whom p = .75. Find the probabilty that Julie socres 70% or lower on a 100-question test.
here what the back of the book says
P(p hat less than or equal to .50) = P(Z less than or equal to -1.155) = .1241
To solve this problem, we need to use the Normal Distribution and Z-scores.
First, let's define some variables:
- p: Probability of correctly answering a question chosen at random from the universe of possible questions.
- p̂: Proportion of correct answers in a sample of size n drawn from the population with population proportion p.
- n: Number of questions on the test.
- Julie's p = 0.75: The probability that Julie answers a question correctly chosen at random from the universe of possible questions.
Now, we want to find the probability that Julie scores 70% or lower on a 100-question test.
To solve this, we need to calculate the z-score for Julie's proportion p̂.
z-score formula: z = (p̂ - p) / sqrt((p * (1 - p)) / n)
Given:
- p = 0.75
- p̂ = 0.70 (70% or lower)
- n = 100
Substituting these values into the formula, we get:
z = (0.70 - 0.75) / sqrt((0.75 * (1 - 0.75)) / 100)
z = (-0.05) / sqrt((0.75 * 0.25) / 100)
z = -1.155
Now, we need to find the probability that the z-score is less than or equal to -1.155 using the Standard Normal Distribution table or a calculator.
From the table, we find that P(Z ≤ -1.155) = 0.1241.
Therefore, the probability that Julie scores 70% or lower on a 100-question test is approximately 0.1241 or 12.41%.