Find the normal approximation for the binomial probability that x = 5, where n = 12 and p = 0.7. Compare this probability to the value of P(x=5) found in Table 2 of Appendix B in your textbook.
We do not have your text book for comparison.
if x=76.9 and s=8.5, compute the test statistic
To find the normal approximation for the binomial probability, you need to use the formula for the mean (μ) and standard deviation (σ) of a binomial distribution.
μ = n * p
σ = sqrt(n * p * (1-p))
In this case, n = 12 and p = 0.7, so we can calculate:
μ = 12 * 0.7 = 8.4
σ = sqrt(12 * 0.7 * (1-0.7)) ≈ 1.44
Next, you can use the normal distribution to approximate the binomial probability. Since x = 5, we want to find P(x = 5). In the normal distribution, we convert this to a standard score (z-score) using the formula:
z = (x - μ) / σ
Substituting the values:
z = (5 - 8.4) / 1.44 ≈ -2.36
Now, we need to use the z-score to find the probability using a standard normal distribution table or a calculator. From the table, you look up the area under the curve to the left of the z-score. In this case, the area is approximately 0.0082 or 0.82%.
Finally, we can compare this probability to the value of P(x=5) found in Table 2 of Appendix B in your textbook. By referring to the table, you can directly read the probability value for P(x=5). If the two values match closely, it demonstrates the accuracy of the normal approximation.