# A. Look for the definitions of the following terms related to hypothesis testing. 1. Null Hypothesis 2. Level of Significance 3. Type I error

Question
Significance tests
A. Look for the definitions of the following terms related to hypothesis testing.
1. Null Hypothesis
2. Level of Significance
3. Type I error

2020-12-29
There are two types of hypotheses: one is the null hypothesis and the other is the alternative hypothesis.
The null hypothesis states there is no significant difference between the population mean and observed mean.
Null hypothesis generally expressed as $$H_{0}$$.
The level of significance means the probability of rejection of the null hypothesis when it is actually true.
A type-I error is rejecting the null hypothesis when it is true.

### Relevant Questions

Hypothesis Testing Review
For each problem below, simply identify the null and alternative hypotheses. Use appropriate notation/symbols. You do not have to run any hypothesis tests, although it's good practice and I'll post answers for all of them.
1) A simple random sample of 44 men from a normally distributed population results in a standard deviation of 10.7 beats per minute. The normal range of pulse rates of adults is typically given as 60 to 100 beats per minute. If the range rule of thumb is applied to that normal range, the result is a standard deviation of 10 beats per minute. Use the sample results with a 0.10 significance level to test the claim that pulse rates of men have a standard deviation equal to 10 beats per minute.
2) In 1997, a survey of 880 households showed that 145 of them use e-mail. Use those sample results to test the claim that more than 15% of households use e-mail. Use a 0.05 significance level.
Hypothesis Testing Review
For each problem below, simply identify the null and alternative hypotheses. Use appropriate notation/symbols. You do not have to run any hypothesis tests, although it's good practice and I'll post answers for all of them.
1) A simple random sample of 44 men from a normally distributed population results in a standard deviation of 10.7 beats per minute. The normal range of pulse rates of adults is typically given as 60 to 100 beats per minute. If the range rule of thumb is applied to that normal range, the result is a standard deviation of 10 beats per minute. Use the sample results with a 0.10 significance level to test the claim that pulse rates of men have a standard deviation equal to 10 beats per minute.
2) In 1997, a survey of 880 households showed that 145 of them use e-mail. Use those sample results to test the claim that more than 15% of households use e-mail. Use a 0.05 significance level.
In there a relationship between confidence intervals and two-tailed hypothesis tests? The answer is yes. Let c be the level of confidence used to construct a confidence interval from sample data. Let * be the level of significance for a two-tailed hypothesis test. The following statement applies to hypothesis tests of the mean:
For a two-tailed hypothesis test with level of significance a and null hypothesis $$H_{0} : \mu = k$$ we reject Ho whenever k falls outside the $$c = 1 — \alpha$$ confidence interval for $$\mu$$ based on the sample data. When A falls within the $$c = 1 — \alpha$$ confidence interval. we do reject $$H_{0}$$.
For a one-tailed hypothesis test with level of significance Ho : $$\mu = k$$ and null hypothesiswe reject Ho whenever A falls outsidethe $$c = 1 — 2\alpha$$ confidence interval for p based on the sample data. When A falls within the $$c = 1 — 2\alpha$$ confidence interval, we do not reject $$H_{0}$$.
A corresponding relationship between confidence intervals and two-tailed hypothesis tests is also valid for other parameters, such as p, $$\mu1 — \mu_2,\ and\ p_{1}, - p_{2}$$.
(a) Consider the hypotheses $$H_{0} : \mu_{1} — \mu_{2} = O\ and\ H_{1} : \mu_{1} — \mu_{2} \neq$$ Suppose a 95% confidence interval for $$\mu_{1} — \mu_{2}$$ contains only positive numbers. Should you reject the null hypothesis when $$\alpha = 0.05$$? Why or why not?
What is the decision at a 0.05 level of significance for each of the following tests?
F(3, 26) = 3.00
Retain or reject the null hypothesis?
F(4, 55) = 2.54
Retain or reject the null hypothesis?
F(4, 30) = 2.72
Retain or reject the null hypothesis?
F(2, 12) = 3.81
Retain or reject the null hypothesis?
A company is marketing a new product they say works better than the traditional test tube. There is so much interest in the product that 30 different labs around the world are testing the claim that this product is actually better. If each study uses an alpha level (alpha) of .10, and if the null hypothesis is true (that the test tube isn't any better that the traditional one), how many of the hypothesis tests would we expect to incorrectly find statistical significance (that is, conclude that the new test tube is better, when it actually isn't)?
As access to computers becomes more and more prevalent, we see the P-value reported in hypothesis testing more frequently. Review the use of the P-value in hypothesis testing. What is the difference between the level of significance of a test and the P-value? Considering both the P-value and level of significance. under what conditions do we reject or fail to reject the null hypothesis?

A random sample of $$n_1 = 14$$ winter days in Denver gave a sample mean pollution index $$x_1 = 43$$.
Previous studies show that $$\sigma_1 = 19$$.
For Englewood (a suburb of Denver), a random sample of $$n_2 = 12$$ winter days gave a sample mean pollution index of $$x_2 = 37$$.
Previous studies show that $$\sigma_2 = 13$$.
Assume the pollution index is normally distributed in both Englewood and Denver.
(a) State the null and alternate hypotheses.
$$H_0:\mu_1=\mu_2.\mu_1>\mu_2$$
$$H_0:\mu_1<\mu_2.\mu_1=\mu_2$$
$$H_0:\mu_1=\mu_2.\mu_1<\mu_2$$
$$H_0:\mu_1=\mu_2.\mu_1\neq\mu_2$$
(b) What sampling distribution will you use? What assumptions are you making? NKS The Student's t. We assume that both population distributions are approximately normal with known standard deviations.
The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.
The standard normal. We assume that both population distributions are approximately normal with known standard deviations.
The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations.
(c) What is the value of the sample test statistic? Compute the corresponding z or t value as appropriate.
(Test the difference $$\mu_1 - \mu_2$$. Round your answer to two decimal places.) NKS (d) Find (or estimate) the P-value. (Round your answer to four decimal places.)
(e) Based on your answers in parts (i)−(iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level \alpha?
At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are not statistically significant.
(f) Interpret your conclusion in the context of the application.
Reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver. (g) Find a 99% confidence interval for
$$\mu_1 - \mu_2$$.
(Round your answers to two decimal places.)
lower limit
upper limit
(h) Explain the meaning of the confidence interval in the context of the problem.
Because the interval contains only positive numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, we can not say that the mean population pollution index for Englewood is different than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains only negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is less than that of Denver.
Suppose you are testing whether the lifetime of 4 different types of lightbulbs. You run 5 experiments each. You get a mean square error of $$\displaystyle{M}{S}_{{{E}}}={1.75}$$ and you reject the null hypothesis. The sample means you obtain for each lightbulb are below:
Which pairs are significantly different at the $$\displaystyle{5}\%$$ significance level? Select all that apply.
For the same data, null hypothesis, and level of significance, if the conclusion is to reject $$H_{0}$$ based on a two-tailed test, do you also reject Ho based on a one-tailed test? Explain.