A popular restaurant downtown says that 15% of diners order the daily special. Consider a random
sample of 108 people who dined at the restaurant. What is the probability that the proportion of diners
who ordered the special differs from the population proportion by less than 2%?
Am I looking to solve this as an fraction?
To calculate the probability that the proportion of diners who ordered the special differs from the population proportion by less than 2%, we need to use the normal approximation to the binomial distribution.
Step 1: Find the standard deviation of the sampling distribution.
The standard deviation of the sampling distribution can be found using the formula:
σ = sqrt(p * (1-p) / n)
where p is the population proportion (0.15) and n is the sample size (108).
σ = sqrt(0.15 * (1-0.15) / 108)
Step 2: Find the standard error.
The standard error is the standard deviation of the sampling distribution divided by the square root of the sample size. Calculate it using the formula:
SE = σ / sqrt(n)
SE = sqrt(0.15 * (1-0.15) / 108) / sqrt(108)
Step 3: Find the difference in proportions.
We want to find the probability that the proportion of diners who ordered the special differs from the population proportion by less than 2%. This means we need to find the range from 0.15 - 0.02 to 0.15 + 0.02. Calculate the difference in proportions:
Difference = (0.15 + 0.02) - (0.15 - 0.02)
Step 4: Calculate the Z-scores.
To use the standard normal distribution table, we need to convert the difference in proportions to a Z-score. Calculate the Z-scores using the formula:
Z1 = (Difference - 0) / SE
Z2 = (Difference - 0) / SE
Step 5: Find the probability.
Using the Z-score table (or a statistical calculator), find the probability associated with each Z-score. Subtract the probability associated with the lower Z-score from the probability associated with the higher Z-score to get the final probability.
Note: It's important to use a standard normal distribution table that provides probabilities for both sides of the distribution. Some tables only provide probabilities for one side, so you need to double the probability found from the table.
By following these steps, you should be able to calculate the probability that the proportion of diners who ordered the special differs from the population proportion by less than 2% using the normal approximation to the binomial distribution.