Question

Calculate the probability of a type ii error if the true mean elongation of drapery yarn is 11.5 kilograms and σ = 0.5 if the situations given below are true. (a) ð’™Ì… = 11.59 𛼠= 0.05 and n = 4 (b) ð’™Ì… = 11.59 𛼠= 0.05 and n = 16 (c) Compare the values for 𛽠calculated in the previous parts. What conclusion can you draw?

Answer 1

To calculate the probability of a Type II error, one must find the critical value, calculate the test statistic, and then determine the probability from the distribution table. Increasing the sample size tends to decrease the probability of making a Type II error.

Step-by-step explanation:

Calculating the probability of a Type II error involves understanding the relationship between the sample size, the significance level, and the actual mean in contrast with the hypothesized mean. The probability of making a Type II error is represented by β, and this error occurs when a false null hypothesis is not rejected.

Calculation Steps:

1. Find the critical value of z or t depending on the sample size and the significance level (α = 0.05).
2. Calculate the test statistic using the actual mean, the hypothesized mean, the standard deviation, and the sample size.
3. Use the test statistic and the distribution table to find the probability of failing to reject the null hypothesis when it is false, which is the probability of a Type II error (β).

For part (a) and (b), we would expect that increasing the sample size (from n=4 to n=16) would reduce the probability of a Type II error since a larger sample size generally leads to a more accurate estimate of the population parameter.

As part (c) asks for a comparison, we can conclude that the probability of a Type II error decreases as the sample size increases, given that other factors like the significance level remain constant.