stats

Researchers studied the mean egg length​ (in millimeters) for a bird population. After taking a random sample of​ eggs, they obtained a​ 95% confidence interval of​ (45,60). What is the value of the margin of​ error?

7.5 mm (middle value or average)

A larger population standard deviation

A larger population standard deviation decreases power.

Type II Error

A Type II Error is made when​ there's not enough evidence to reject the null​ hypothesis, but the null hypothesis is not true.

Which of the following is a correct explanation of what a confidence interval​ is?

A confidence interval is a range of values used to estimate the true value of a population parameter. The confidence level is the probability the interval actually contains the population​ parameter, assuming that the estimation process is repeated a large number of times.

​(a) A larger predicted difference between the means of the populations

A larger predicted difference between the means of the populations increases power

A larger sample size

A larger sample size increases power.

Suppose that you take 1000 simple random samples from a population and​ that, for each​ sample, you obtain a​ 95% confidence interval for an unknown parameter. Approximately how many of those confidence intervals will contain the value of the unknown​ parameter?

Approximately 950 confidence intervals will contain the value of the unknown parameter.

A researcher developing scanners to search for hidden weapons at airports has concluded that a new scanner is significantly better than the current scanner. He made his decision based on a test using α=0.05. Would he have made the same decision at α=0.20? How about α=0.001?

His decision would have been the same for α=0.20 but may have been different for α=0.001.

Which of the following are mistakes that can be made in a hypothesis​ test? I. H0 is​ true, and we reject it. II. H0 is​ true, and we fail to reject it. III.H0 is​ false, and we fail to reject it

I and III

Jan performed a study and obtained a​ p-value of 1.24. What conclusion should Jan​ make?

She made an error since it is not possible to get a​ p-value of 1.24.

A study was conducted based on a sample size of 30 individuals. The​ p-value was 0.10. Suppose a researcher conducted another study by taking a random sample of 50 individuals from the same population. Suppose they obtained the same sample mean as in the first study with a sample size of 30.​ (Also assume the population standard deviation is the same for both​ studies.) Which of the following is​ true?

The​ p-value would be smaller for the second study.

What happens to the power of a hypothesis test if the sample size is increased without changing the significance​ level?

The power of the test increases because the standard deviation of the sampling distribution of the mean decreases. That means that the null hypothesis is rejected for smaller differences between the sample mean and the actual mean.

Researchers can make the results statistically significant by increasing the sample size even if the difference between the sample mean and hypothesized value of the population mean is very small.

The statement is true.

A​ p-value is the probability of accepting the null hypothesis.

This statement is false. We never accept the null hypothesis no matter what the​ p-value is. A​ p-value is the probability of observing a sample mean​ (for example) that we did or something more unusual just by chance if the null hypothesis is true.

Using a more extreme significance level​ (for example, 0.01 instead of​ 0.05)

Using a more extreme significance level decreases power.

The​ P-value for a hypothesis test is 0.073. For each of the following significance​ levels, decide whether the null hypothesis should be rejected.

.05 - do not reject the null hypothesis because the​ P-value is greater than the significance level. 0.10 - Reject the null hypothesis because the​P-value is equal to or less than the significance level.

A​ p-value is the probability​ _____________.

A​ p-value is the probability of observing the actual​ result, a sample​ mean, for​ example, or something more unusual just by chance if the null hypothesis is true.

Using a​ two-tailed test instead of a​ one-tailed test

Using a​ two-tailed test instead of a​ one-tailed test decreases power.

Interpret the confidence interval

We can be 95​% confident that the mean duration of​ imprisonment, μ​, of all political prisoners with chronic PTSD is somewhere between 18.318.3 months and 46.146.1 months.

When are conclusions said to be​ "statistically significant"?

When the​ p-value is less than a given significance level

Suppose​ that, in a hypothesis​ test, the null hypothesis is in fact false

​No, it is not possible. A Type I error is rejecting the null hypothesis when it is in fact true. ​Yes, it is possible. A Type II error is not rejecting the null hypothesis when it is in fact false.

errors

A Type I error is rejecting a true null​ hypothesis, whose probability is denoted α. A Type II error is not rejecting a false null​ hypothesis, whose probability is denoted β.

The​ P-value is .363 test at the 10​% significance level.

Do not reject the null hypothesis. The data do not provide sufficient evidence to conclude that the mean is not equal to 34.

Researchers conducted a study and obtained a​ p-value of 0.75. Based on this​ p-value, what conclusion should the researchers​ draw?

Fail to reject the null hypothesis but do not accept the null hypothesis as true either.

Power

Power is the probability that a hypothesis test will correctly reject a false null hypothesis.

It is recommended that adults get 8 hours of sleep each night. A researcher hypothesized college students got less than the recommended number of hours of sleep each​ night, on average. After randomly sampling a number of college​ students, the researcher calculated a sample mean of 7.95 hours and obtained a​ p-value of 0.0001 from a​ one-sample t-test. Why is the​ p-value so small when the difference between the sample mean and the hypothesized value of the population mean is only 0.05 hours​ (3 minutes)?

The sample size was very large.

The value obtained for the test​ statistic, z, in a​ one-mean z-test is given. Whether the test is two​ tailed, left​ tailed, or right tailed is also specified. For parts​ (a) and​ (b), determine the​ P-value and decide​ whether, at the 5​% significance​ level, the data provide sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis.

The test statistic in a two-tailed test is z=−1.08. The​ P-value is 0.280 ​(Round to three decimal places as​ needed.) At the 5​% significance​ level, the data does not provide sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis because the obtained​ P-value is greater than the significance level. b. The test statistic in a two-tailed test is z=0.54. The​ P-value is . 589 ​(Round to three decimal places as​ needed.) At the 5​% significance​ level, the data does not provide sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis because the obtained​ P-value is greater than the significance level.

The value of a statistic used to estimate a parameter is called

The value of a statistic used to estimate a parameter is called​ a(n) point estimate (mean) of the parameter.

It is recommended that adults get 8 hours of sleep each night. A researcher hypothesized college students got less than the recommended number of hours of sleep each​ night, on average. The researcher randomly sampled 20 college students and found no evidence to reject the null hypothesis at the​ 5% significance level. What is true regarding the​ p-value from this hypothesis​ test?

The​ p-value must have been greater than 0.05.

Suppose increasing the sample size will not change the sample mean or the standard deviation. What will happen to the​ p-value by increasing the sample​ size?

The​ p-value will decrease.

It is recommended that adults get 8 hours of sleep each night. A researcher hypothesized college students got less than the recommended number of hours of sleep each​ night, on average. The researcher randomly sampled 20 college students and obtained a​ p-value of 0.10. Suppose the researcher sampled more college students and that the sample mean and sample standard deviation stayed the same. Would the​ p-value be​ lower, be​ higher, or stay the​ same?

The​ p-value would be lower compared to the​ p-value from the sample with 20 college students.