Consider the hypothesis test H0: μ1=μ2 against H1: μ1 ≠μ2 with known variances σ1=10 and σ2=5 Su

Consider the hypothesis test H0: μ1=μ2 against H1: μ1 ≠μ2...

Consider the hypothesis test $H_{0}: \mu_{1}=\mu_{2}$ against $H_{1}: \mu_{1} \neq \mu_{2}$ with known variances $\sigma_{1}=10$ and $\sigma_{2}=5$ Suppose that sample sizes $n_{1}=10$ and $n_{2}=15$ and that $\bar{x}_{1}=4.7$ and $\bar{x}_{2}=7.8 .$ Use $\alpha=0.05$. (a) Test the hypothesis and find the $P$ -value. (b) Explain how the test could be conducted with a confidence interval. (c) What is the power of the test in part (a) for a true difference in means of $3 ?$ (d) Assuming equal sample sizes, what sample size should be used to obtain $\beta=0.05$ if the true difference in means is $3 ?$ Assume that $\alpha=0.05 .$

Consider the hypothesis test $H_{0}: \mu_{1}=\mu_{2}$ against $H_{1}: \mu_{1} \neq \mu_{2}$ with known variances $\sigma_{1}=10$ and $\sigma_{2}=5$ Suppose that sample sizes $n_{1}=10$ and $n_{2}=15$ and that $\bar{x}_{1}=4.7$ and $\bar{x}_{2}=7.8 .$ Use $\alpha=0.05$.
(a) Test the hypothesis and find the $P$ -value.
(b) Explain how the test could be conducted with a confidence interval.
(c) What is the power of the test in part (a) for a true difference in means of $3 ?$
(d) Assuming equal sample sizes, what sample size should be used to obtain $\beta=0.05$ if the true difference in means is $3 ?$ Assume that $\alpha=0.05 .$

Key Concepts

Sample Size Determination

Determining the appropriate sample size involves balancing the risks of Type I and Type II errors while ensuring that the study is adequately powered to detect a meaningful difference between groups. Sample size calculations take into account the desired significance level, the effect size of interest, and the variability in the data, providing a guideline to achieve a specified power level.

Two-Sample Hypothesis Testing

This concept involves comparing the means of two distinct populations to determine if there is a statistically significant difference between them. The test uses sample data and often involves calculating a test statistic under an assumed null hypothesis (that the population means are equal) versus an alternative hypothesis (that they are different), taking into account the variability from each population.

Significance Level and p-value

The significance level, denoted by ?, is the threshold probability for rejecting the null hypothesis when it is true (Type I error). The p-value is the probability, under the null hypothesis, of obtaining a test statistic as extreme as or more extreme than the observed value. It provides a measure to decide whether to reject the null hypothesis based on how small the p-value is relative to the predefined ?.

Confidence Interval in Hypothesis Testing

A confidence interval provides a range of plausible values for an unknown population parameter, such as the difference in means. If the hypothesized difference (e.g., zero) does not fall within this interval, the null hypothesis can be rejected at a corresponding confidence level. This method offers a dual perspective to hypothesis testing by quantifying the estimate’s precision.

Statistical Power

Statistical power is the probability that a test will correctly reject a false null hypothesis (i.e., detect a true effect when it exists). Power analysis helps to assess the sensitivity of the test to detect a specified effect size and is complementary to the considerations of Type II error probability (?). It is crucial for designing experiments, particularly in determining appropriate sample sizes.

Transcript

00:02 Hi guys, this problem from the given information, the null hypothesis, each node, mu1 equals mu2, and the alternative hypothesis, mu1 does not equal mu2, okay? and we have variance 1 is 10, it's, and variance 2, it's 5, and we have sample size set, that n1 equals 10 n2 equals 15 and sample means are x1 dash equal 4 .7 and x2 dash equals 7 .8 okay and we have the level of significance alpha equals 0 .05 okay now the quantity of interest is a difference in means mu 1 minus mu 2 so know that delta node equal zero use the following formula to compute the test statistics such that z node equals x1 dash minus x2 dash so it's 4 .7 minus 7 .8 minus delta node which is 0 over the square root of sigma 1 squared over n1 which is 10 over 10 plus 5 squared over 15 okay so it's negative 0 .91 okay from the standard normal table the critical value corresponding to 0 .05 level of significance is 1 .96 so a decision rule we will reject the null hypothesis if the test statistics value is less than the critical value.

02:07 Okay.

02:08 Or is the test statistic value is greater than the critical value.

02:12 And here we see that negative 0 .91 is more than negative 1 .96.

02:21 Okay.

02:22 That means the test statistic value is greater than the critical value.

02:25 So we fail to reject the null hypothesis.

02:29 Okay.

02:30 Okay.

02:30 Now we will compute the p value.

02:33 Okay.

02:33 So the p value equals 2 times 1 minus phi of modulus of the node.

02:48 So this is 2 times 1 minus 0 .81 86...