Test a claim p=0.5 with the significance level of 0.05 and n=49 calculate beta when the true probability porportion is 0.7.
To calculate the beta (β) in hypothesis testing, we need to understand the concept of the power of a test (1 - β). The power of a test measures the probability of correctly rejecting the null hypothesis when it is false. In other words, the power represents the ability of a test to detect a true effect.
To calculate the beta, we first need to determine the critical value from the significance level. Since the significance level is 0.05, we'll use a z-score corresponding to that level.
The critical value (z_c) can be found using a standard normal distribution table or a statistical software tool. For a two-tailed test at a significance level of 0.05, the critical value is approximately ±1.96.
Next, we need to calculate the standard error. The standard error (SE) of a sample proportion (p̂) can be calculated using the formula:
SE = √[(p̂ * (1 - p̂)) / n]
In this case, p̂ is the claim value, which is 0.5, and n is the sample size, which is 49. So we have:
SE = √[(0.5 * (1 - 0.5)) / 49]
SE = √[0.25 / 49]
SE ≈ 0.112
Now, we can calculate the difference between the claim value (p) and the true probability proportion (p_1):
d = p - p_1
d = 0.5 - 0.7 = -0.2
Next, we calculate the test statistic (z):
z = (d - 0) / SE
z = (-0.2 - 0) / 0.112
z ≈ -1.79
Finally, we can calculate the beta (β) using the formula:
β = P(z ≤ z_c + z | p = p_1)
where P(z ≤ z_c + z | p = p_1) is the probability that the test statistic falls in the non-rejection region when the true probability proportion is p_1.
Since this is a two-tailed test, we need to find the areas under the normal curve to the left and right of the test statistic (z) and compare them to the critical value (z_c). We can use a statistical software tool or a standard normal distribution table to find these probabilities.
Once we have those probabilities, we can calculate β as follows:
β = P(z ≤ z_c + z | p = p_1) + P(z ≥ -z_c + z | p = p_1)
Note that in this case, since we have a negative z value, we need to use the probability associated with the right tail (P(z ≥ -z_c + z | p = p_1)).
By substituting the values we obtained earlier:
β ≈ P(z ≤ 1.96 - 1.79 | p = 0.7) + P(z ≥ -1.96 - 1.79 | p = 0.7)
Now, you can use statistical software or a standard normal distribution table to find these probabilities. The calculated result will provide you with the value of beta (β) for this specific scenario.