When considering a Type I error: OA. The probability of occurrence is denoted by Beta (β) OB. The power of a test determines the probability of a Type I error OC. Should be set so that investigators always reject the null hypothesis. OD. Such an error is made when we reject Ho given the Ho is true
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Step 1: A Type I error is rejecting the null hypothesis when it is actually true. Show more…
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Assume that you are using a significance level of $\alpha=0.05$ to test the claim that $p>0.5$ and that your sample is a simple random sample of size $n=64$ a. Assuming that the true population proportion is $0.65,$ find the power of the test, which is the probability of rejecting the null hypothesis when it is false. (In the procedure below, we refer to $p=0.5$ as the "assumed" value, because it is assumed in the null hypothesis; we refer to $p=0.65$ as the "alternative" value, because it is the value of the population proportion used as an alternative to 0.5.) Use the following procedure and see the figure below. (FIGURE CAN'T COPY) Step 1: $\quad$ Using the significance level, find the critical $z$ value(s). (For a right-tailed test, there is a single critical $z$ value that is positive; for a left-tailed test, there is a single critical value of $z$ that is negative; and a two-tailed test will have a critical $z$ value that is negative along with another critical $z$ value that is positive.) Step $2: \quad$ In the expression for the test statistic below, substitute the assumed value of $p$ (used in the null hypothesis). Evaluate $1-p$ and substitute that result for the en- try of $q$. Also substitute the critical value(s) for $z$. Then solve for the sample statistic $\hat{p} .$ (If the test is two-tailed, substitute the critical value of $z$ that is positive, then solve for the sample statistic $\hat{p}$. Next, substitute the critical value of $z$ that is negative, and solve for the sample statistic $\hat{p} .$ A two-tailed test should therefore result in two different values of $\hat{p} .$ The resulting value(s) of $\hat{p}$ separate the region(s) where the null hypothesis is rejected from the region where we fail to reject the null hypothesis. $$z=\frac{\hat{p}-p}{\sqrt{\frac{p q}{n}}}$$ Step 3: $\quad$ The calculation of power requires a specific value of $p$ that is to be used as an alternative to the value assumed in the null hypothesis. Identify this alternative value of $p$ (not the value used in the null hypothesis), draw a normal curve with this alternative value at the center, and plot the value(s) of $\hat{p}$ found in Step 2 Step 4: $\quad$ Refer to the graph in Step $3,$ and find the area of the new critical region bounded by the value(s) of $\hat{p}$ found in Step 2 . (Caution: When evaluating $\sqrt{p q / n},$ be sure to use the alternative value of $p,$ not the value of $p$ used for the null hypothesis.) This is the probability of rejecting the null hypothesis, given that the alternative value of $p$ is the true value of the population proportion. Because this is the probability of rejecting the false null hypothesis, it is the power of the test. b. Find $\beta,$ which is the probability of failing to reject the false null hypothesis. The value of $\beta$ is easily determined by finding the complement of the power.
Kari H.
For a hypothesis tot with a specified significance level $\alpha$, the probability of a type I error is $\alpha,$ whereas the probability $\beta$ of a type II error depends on the particular value of $p$ that is used as an alternative to the null hypothesis. a. Using an alternative hypothesis of $p<0.4,$ a sample size of $n=50$, and assuming that the true value of $p$ is $0.25,$ find the power of the test. See Exercise 47 in Section $8-2 .$ (Hint: Use the values $p=0.25 \text { and } p q / n=(0.25)(0.75) / 50 .)$ b. Find the value of $\beta,$ the probability of making a type II error. c. Given the conditions cited in part (a), what do the results indicate about the effectiveness of the hypothesis test?
Hypothesis Testing
Testing a Claim About a Proportion
1. A Type I error is committed when __________. the null hypothesis is true and it is not rejected the null hypothesis is true and it is rejected the null hypothesis is false and it is not rejected the null hypothesis is false and it is rejected any incorrect rejection is made 2. A researcher is testing a hypothesis of a single mean. The critical z value for α = .05 and a two tailed test is ±1.96. The observed z value from sample data is minus 1.85. The decision made by the researcher based on this information is to _____ the null hypothesis. reject fail to reject redefine change the alternate hypothesis into restate 3. If α is the probability of committing a Type I error and β is the probability of committing a Type II error in a hypothesis test, the power of the test is given by __________.
Evelyn C.
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