I am having so much trouble calculating Type 2 error in simple questions, can someone pleaseeeee help me by explaining what to do?
Let ƒÊ denote the true average diameter for bearings of a certain type. A test of Ho: ƒÊ = 0.5 versus Ha: ƒÊ �‚ 0.5 will be based on a sample of n bearings. The diameter distribution is believed to be normal. Determine the value of ƒÀ in each of the following cases. (Give the answers to two decimal places.)
n = 10, ƒ¿ = .01, ƒÐ = 0.02, ƒÊ = 0.52
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To calculate the Type 2 error in this case, we need to understand the concept of Type 2 error and the basic steps involved in its calculation.
Type 2 error refers to the situation when we fail to reject a null hypothesis (Ho) even though it is false. In other words, it is the probability of accepting the null hypothesis when the alternative hypothesis (Ha) is true.
To calculate the Type 2 error, we need to determine the critical value (ƒÀ), which defines the region of acceptance for the null hypothesis. The critical value is determined based on the significance level (ƒ¿), the standard deviation (ƒÐ), the true average (ƒÊ), and the sample size (n).
In this case, we are given the following values:
- Sample size (n) = 10
- Significance level (ƒ¿) = .01
- Standard Deviation (ƒÐ) = 0.02
- True average (ƒÊ) = 0.52
To calculate the critical value (ƒÀ), we need to determine the value of the test statistic that corresponds to the given significance level. Since the diameter distribution is believed to be normal, we can use the standard normal distribution table or a statistical calculator to find this value.
To find the critical value, we need to follow these steps:
Step 1: Calculate the standard error (SE) of the sampling distribution.
SE = ƒÐ / √n
Step 2: Calculate the z-score corresponding to the significance level (ƒ¿).
z-score = invNorm(ƒ¿)
Step 3: Calculate the difference between the true average (ƒÊ) and the hypothesized average under the null hypothesis (Ho).
ƒ�= |ƒÊ - 0.5|
Step 4: Calculate the critical value by multiplying the standard error (SE) by the z-score and adding it to the hypothesized average under the null hypothesis (Ho).
ƒÀ = 0.5 + (z-score * SE)
Using these steps and the given values, we can now calculate the Type 2 error for this specific case.
1. Calculate the standard error (SE):
SE = 0.02 / √10 ≈ 0.00632
2. Calculate the z-score corresponding to the significance level (ƒ¿):
z-score = invNorm(0.01) ≈ -2.33
3. Calculate the difference between the true average (ƒÊ) and the hypothesized average under the null hypothesis (Ho):
ƒ�= |0.52 - 0.5| = 0.02
4. Calculate the critical value (ƒÀ) by multiplying the standard error (SE) by the z-score and adding it to the hypothesized average under the null hypothesis (Ho):
ƒÀ = 0.5 + (-2.33 * 0.00632) ≈ 0.4862
Therefore, the value of ƒÀ in this case is approximately 0.4862.
By following these steps, you can calculate the Type 2 error in similar hypothesis testing cases. Just remember to gather the necessary information like significance level, standard deviation, true average, and sample size, and apply the steps accordingly.