3818 module 10

A researcher looking for evidence of extrasensory perception (ESP) tests 10001000 subjects. Nine of these subjects do significantly better (𝑃<0.01)(P<0.01) than random guessing. (a) Nine subjects may seem like a lot of people, but can you conclude that these nine people have ESP? Select the appropriate statement that explains whether or not it is proper to conclude that these nine people have ESP. (b) Select the appropriate statement that describes what the researcher should now do to test whether any of these nine subjects have ESP.

(a) No. Since the tests were performed at the 1%1% significance level, as many as 1010 subjects may have done significantly better than random guessing. (b) He should design a new test and draw new data

probability of type 2 error

1 - power

probability of type 1 error type 2 error

1 = alpha 2 = beta

given information, find the upper and lower bounds at a 0.05 significance level find the probability of obtaining a sample mean with the critical region for a specific value of alternative u

1) use mean +/- (z * std/sqrt n) *not sure why they assumed 95% confidence interval 2) power is the probability of obtaining a mean less than or greater than the confidence interval bounds - use the z scores to find the probability of rejecting the null hypothesis the rainfall test was not powerful because it was asked to calculate a tiny difference where the standard error included was included in this range

When a study involves a test about a mean, data should always be scrutinized for possible outliers or heavily skewed data. The 𝑡t test and the 𝑡t interval rely on certain assumptions. Which of the given statements is true?

1. As long as the sample size is at least 15 and there are no outliers among the data, we can use a 𝑡t test. 2. The most important aspect for validity is a simple random sample from the population of interest. 3. As long as the sample size is at least 40, we can use the 𝑡t test even if the data exhibits skewness.

A test of significance was performed using 𝛼=5% . The 𝑃P‑value was 4% and the power was 85%. What is the probability of a Type II error for this test?

15% The power is the probability that we accurately accept the null hypothesis when it is true. Therefore, the odds of rejecting it when its false (type 2 error) 1 - power = 15%

If a researcher wants to compute a 90% confidence interval from a population with a known standard deviation of 5 units and have a margin of error of no more than 1 unit, what is the smallest sample size that can be used?

Population standard deviation = = 5 Margin of error = E = 1 At 90% confidence level the z is 1.645 sample size = n = [Z/2* / E] 2 n = [1.645 * 5 / 1 ]2 n = 67.65 answer: 68

Your sample size calculation to control margin of error for a confidence interval had a result of 72.5. You should get a sample of:

The calcualtions for sample size is rounded up to next integer. 73 individuals

If a research team increases the sample size for a study, the power of their statistical tests will:

increase

What is the probability that a test will reject 𝐻0 at a chosen significance level when a specified alternative value of the parameter is true?

power

A college administrator wanted to make all freshmen live on campus, because he thought that this would lead to better study habits and better grades. He commissioned a study that found a P‑value less than 0.05 for the difference in GPAs between freshmen who lived on campus and freshmen who commuted. The average GPA for the random sample of freshmen who lived on campus was 2.53, and the average GPA for the random sample of freshmen who commuted was 2.51. When he presented his findings, his idea was rejected.

the findings were not practically significant

Power of a statistical test

the probability of rejecting the null hypothesis when it is false

A NHANES report gives data for 654654 women aged 20-2920-29 years. The mean BMI of these 654654 women was 𝑥¯=26.8. We treated these data as an SRS from a normally distributed population with standard deviation 𝜎=7.5. (a) Suppose that we had an SRS of just 100 young women. What would be the margin of error for 95% confidence? (b) Find the margins of error for SRSs of 400400 young women and 16001600 young women. (c) Compare the margins of error for 100, 400, and 1600 women. How does increasing the sample size change the margin of error of a confidence interval when the confidence level and population standard deviation remain the same? The larger the sample size, the larger margin of error of a confidence interval.

z-score for a 95% confidence interval is 1.96 margin of error = z * (sd/sqrt n) (a) 1.47 (b) n=400, m=0.735 n=1600, m=0.3675 (c) False. The margin of error is halfed each time 𝑛n quadruples.

(a) You could get higher power against the same alternative with the same 𝛼 by changing the number of measurements you make. Select the appropriate procedure that you should do to increase power. (b) Suppose you decide to use 𝛼=0.10α=0.10 in place of 𝛼=0.05,α=0.05, with no other changes in the test. Select the appropriate statement that describes what will happen to power. (c) Suppose you shift your interest to the alternative 𝜇=225 with no other changes. Select the appropriate statement that describes what will happen to power.

(a) the larger the sample size, the higher the power so "we should make more measurements" (b) There is a trade-off between the significance level and power: the lower the significance level, the lower the power "the power will increase" (c) The power will increase, because this alternative is more likely to be detected.

if everything else remains constant, the following will cause in increase in power

- increasing the sample size - increasing the significance level - increasing the different the test is required to detect

students who admitted texting while driving 1) Why is this estimate likely to be biased? 2) baised high or low? 3) Does the margin of error of a 95% confidence interval allow for this bias?

1) Many students might be reluctant to confess that they had participated in this risky behavior 2) low 3) No, the margin of error only covers random sampling errors.

A randomly sampled group of patients at a major U.S. regional hospital became part of a nutrition study on dietary habits. Part of the study consisted of a 50‑question survey asking about types of foods consumed. Each question was scored on a scale from one: most unhealthy behavior, to five: most healthy behavior. The answers were summed and averaged. The population of interest is the patients at the regional hospital. The current survey was implemented after patients were subjected to this education, and 100 patients were included in the sample. The 𝑡t test for the hypotheses 𝐻0:𝜇=2.9 versus 𝐻𝛼:𝜇>2.9 was 𝑡=2.88.

The 𝑃‑value is significant at: a = 0.01, 0.1, 0.5 the greater the value of t, the greater the evidence against the null hypothesis *check chegg bookmarks

What is the probability that we reject 𝐻0 when 𝐻0 is true?

Type 1 error

What is the probability that we fail to reject 𝐻0 when 𝐻a is true?

Type 2 error

true statements about the power of a hypothesis test (a) If the sample size increases and everything else remains the same, power increases. (b) If the population standard deviation increases and everything else remains the same, the power increases. (c) If the significance level, 𝛼, increases and everything else remains the same, power increases. (d) If the probability of a type II error, 𝛽, increases and everything else remains the same, power increases. (e) If the hypothesis test is used to detect a smaller difference, power increases.

a) true b) False, there is no effect of standard deviation on power. c) False, If the hypothesis test is used to detect a smaller difference, power increases. d) False- power is equal to one minus beta, the power of a test will get smaller as beta gets bigger. e) true

A survey was sent to 1000 randomly selected registered drivers in Atlanta asking for the average number of miles driven to and from work each weekday. A 95% confidence interval was computed using the responses obtained from the 720 drivers who returned the survey. The margin of error for the interval was reported to be 5 miles.

error due to random sampling

wildlife in several west african game preserves (a) Examine the distribution of the data. Select the feature of the distribution that throws doubt on the validity of statistical inference? (b) Plot the percents against year. Select the best description of the trend in this time series. (c) Select the answer that best explains why a trend over time casts doubt on the condition that years 1971 to 1999 can be treated as an SRS from a larger population of years.

c) when the data show a trend over time, the sample cannot be treated as if it comes from a single population

Nina decreases the confidence level to 90%? Nina increases the confidence level to 99%? Nina decreases the sample size to 31 locations? Nina increases the sample size to 55 locations? the margin of error will

decrease increase increase decrease