Final answer:
The probability that the proportion of defective bottles in a sample of 730 bottles would differ from the population proportion by less than 3% is approximately 0.0697.
Step-by-step explanation:
To calculate the probability that the proportion of defective bottles in a sample of 730 bottles would differ from the population proportion by less than 3%, we can use the normal approximation to the binomial distribution.
Calculate the Standard Deviation
First, we need to calculate the standard deviation (σ) using the formula:
σ = √(p*(1-p)/n)
Where:
p = population proportion of defective bottles (0.33)
n = sample size (730)
σ = √(0.33(1-0.33)/730)
= √(0.330.67/730)
= √(0.2211/730)
= 0.01742
Calculate the Margin of Error
The margin of error is given by multiplying the standard deviation by the z-score corresponding to the desired level of confidence.
Since we want to find the probability that the proportion differs by less than 3%, we can use a z-score that corresponds to a 97% confidence level, which is approximately 2 standard deviations from the mean.
Margin of Error = z * σ
= 2 * 0.01742
=0.03484
Calculate Lower and Upper Bound for Proportion
Next, we calculate the lower and upper bounds for the proportion of defective bottles in the sample:
Lower Bound = p - Margin of Error
= 0.33 - 0.03484
= 0.29516
Upper Bound = p + Margin of Error
= 0.33 + 0.03484
≈ 0.36484
Calculate Probability
Finally, we can calculate the probability that the proportion of defective bottles in a sample of 730 bottles would differ from the population proportion by less than 3% using these bounds.
Probability = Upper Bound - Lower Bound
≈ 0.36484 - 0.29516
≈ 0.0697
Therefore, the probability that the proportion of defective bottles in a sample of 730 bottles would differ from the population proportion by less than 3% is approximately 0.0697.