A sample survey interviews an SRS of 115 college women. Suppose that 68% of all college women have been on a diet within the last 12 months. What is the probability that 76% or more of the women in the sample have been on a diet?
To find the probability that 76% or more of the women in the sample have been on a diet, we can use the binomial distribution.
Let's define the random variable X as the number of women in the sample who have been on a diet. X follows a binomial distribution with parameters n = 115 (sample size) and p = 0.68 (probability of success, i.e. being on a diet).
To calculate the probability, we need to find the cumulative probability from 76% (0.76) up to 100% (1). We can use the cumulative distribution function (CDF) of the binomial distribution to do this.
P(X ≥ 0.76 * 115) = 1 - P(X ≤ 0.76 * 115 - 1)
P(X ≥ 87.4) = 1 - P(X ≤ 87)
Using a binomial distribution table or a statistical software, we can find the probability:
P(X ≥ 87) = 1 - ∑ (from k = 0 to 86) [C(115, k) * p^k * (1 - p)^(n - k)]
Alternatively, we can use a calculator or software with the binomial cumulative distribution function (CDF) to directly calculate the probability:
P(X ≥ 87) = 1 - binomcdf(115, 0.68, 86)
After calculating this probability, you will have the answer to the question.
To find the probability that 76% or more of the women in the sample have been on a diet, we need to use the concept of binomial distribution. Let's break down the steps to calculate it:
Step 1: Identify the parameters:
- Number of trials (n): the size of the sample, which is 115 women.
- Probability of success (p): the proportion of college women who have been on a diet, which is given as 0.68.
- Probability of failure (q): the complement of the probability of success, which is 1 - p = 1 - 0.68 = 0.32.
Step 2: Set up the binomial probability formula:
The probability mass function for the binomial distribution is given by the formula:
P(X = k) = nCk * p^k * q^(n-k)
Where X is the random variable representing the number of college women in the sample who have been on a diet, k is the number of successes, nCk denotes the binomial coefficient (also known as combinations), and p and q are the probabilities of success and failure, respectively.
Step 3: Calculate the probability of 76% or more women being on a diet:
To find the probability of 76% or more women being on a diet, we need to calculate the sum of probabilities for all values from 76% to 100%. Since calculating the exact values can be tedious, we can use software or a calculator to do the computations.
Using software like R or Python with the binomial distribution function, we can calculate this probability directly. For example, in R, you can use the `pbinom` function:
p_value <- 1 - pbinom(0.76 * 115, 115, 0.68)
This calculates the cumulative probability of having 76% or more successes in 115 trials with a success probability of 0.68.
By evaluating this expression, we obtain the probability that 76% or more of the women in the sample have been on a diet.