According to a Wakefield Research survey of adult women, 50% of the women said that they had
tried five or more diets in their lifetime (USA TODAY, June 21, 2011). Suppose that this result is
true for the current population of adult women. A random sample of 13 adult women is selected.
Use the binomial probabilities table or technology to find the probability that the number of
women in this sample of 13 who had tried five or more diets in their lifetime is
P(5 to 7)=

To solve this problem, we will use the **binomial distribution** formula. The binomial distribution is defined as:

\[
P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}
\]

Where:
- \(n = 13\) is the number of trials (the sample of women).
- \(p = 0.50\) is the probability of success (a woman having tried 5 or more diets).
- \(k\) is the number of successes (we want to calculate the probability for \(k\) being 5, 6, and 7).
- \(\binom{n}{k}\) is the binomial coefficient, which represents the number of ways to choose \(k\) successes from \(n\) trials.

We want to find:

\[
P(5 \leq X \leq 7) = P(X = 5) + P(X = 6) + P(X = 7)
\]

I will now calculate these probabilities.The probability that the number of women in a sample of 13 who have tried five or more diets is between 5 and 7 (inclusive) is approximately **0.576**.

Would you like a detailed breakdown of the calculation process or have any other questions?

Here are 5 related questions you might consider:
1. What is the probability that exactly 5 women have tried five or more diets?
2. What is the probability that fewer than 5 women have tried five or more diets?
3. What is the probability that more than 7 women have tried five or more diets?
4. How does the probability change if the sample size increases to 20 women?
5. What is the expected number of women who have tried five or more diets in a sample of 13?

**Tip:** In a binomial distribution, the mean can be found using \( \mu = n \cdot p \) and the variance with \( \sigma^2 = n \cdot p \cdot (1 - p) \).

According to a Wakefield Research survey of adult women, 50% of the women said that they had tried five or more diets in their lifetime (USA TODAY, June 21, 2011). Suppose that this result is true for the current population of adult women. A random sample of 13 adult women is selected. Use the binomial probabilities table or technology to find the probability that the number of women in this sample of 13 who had tried five or more diets in their lifetime is P(5 to 7).

Explore how to calculate the probability that between 5 to 7 women in a sample of 13 have tried five or more diets using the binomial distribution. This problem applies key concepts from probability and statistics, with detailed formulas and step-by-step solutions. Suitable for high school students.

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