A hypothesis test for a population mean is to be performed at the 5% level of significance. The population standard deviation is known. The hypotheses are
H0: ƒÊ = 80
Ha: ƒÊ �‚ 80.
A 95% confidence interval for ƒÊ is also constructed. True or false, if 80 lies within the 95% confidence interval, then the null hypothesis will be rejected?
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the answer is false
I think the null hypothesis should not be rejected..
Can I ask for further, simple explanation of this question, ASAP?? Thanks alot
ƒÊ = mu
Sure! Let's break down the question and explain why the answer is false.
In hypothesis testing, we compare the sample data to a predefined null hypothesis (H0) and an alternative hypothesis (Ha). The null hypothesis states there is no significant difference between the population mean (μ) and a specific value, which in this case is 80.
The alternative hypothesis, Ha, states there is a significant difference. In this case, it specifically says "not equal to" 80, indicating a two-tailed test.
The question asks whether rejecting the null hypothesis depends on whether the value of 80 lies within a 95% confidence interval for the population mean. To answer this, we need to understand the relationship between hypothesis testing and confidence intervals.
When performing a hypothesis test, we set a significance level, often denoted as α, which represents the probability of making a Type I error (incorrectly rejecting a true null hypothesis). In this question, the significance level is 5%, or α = 0.05.
A confidence interval, on the other hand, estimates the range within which the population parameter lies with a certain level of confidence. In this case, a 95% confidence interval is constructed, meaning we can be 95% confident that the interval contains the true population mean.
Now, if the null hypothesis is true (μ = 80), there is a 5% chance of observing a sample mean that falls outside the confidence interval due to sampling variability. This means that in 5% of cases, the sample mean may fall outside the 95% confidence interval, even though the true population mean is indeed 80.
Therefore, the fact that 80 lies within the 95% confidence interval suggests that it is a plausible value for the population mean. Hence, we would not be able to reject the null hypothesis.
To summarize, if the value of 80 lies within the 95% confidence interval, it implies that the null hypothesis would not be rejected. Therefore, the correct answer is false.