Confidence Interval to Significance Test

A botanist collected one leaf at random from each of 10 randomly selected mature maple trees of the same species. The mean and the standard deviation of the surface areas for the 10 leaves in the sample were computed.Assume the distribution of surface areas of maple leaves is normal. What is the appropriate method for constructing a one-sample confidence interval to estimate the population mean surface area of the species of maple leaves, and why is the method appropriate?

• The t-interval is appropriate, because the population standard deviation is not known.

• The t-interval is appropriate, because the t-interval is narrower than the z-interval.

• The z-interval is appropriate, because the z-interval is narrower than the t-interval.

• The z-interval is appropriate, because the central limit theorem applies.

• The z-interval is appropriate, because the sample standard deviation is known.

A state educational agency was concerned that the salaries of public school teachers in one region of the state,region A, were higher than the salaries in another region of the state, region B. The agency took two independent random samples of salaries of public school teachers, one from region A and one from region B. The data are summarized in the table below. Assuming all conditions for inference are met, do the data provide convincing statistical evidence that the salaries of public school teachers in region A are, on average, greater than the salaries of public school teachers in region B?

• Yes, there is evidence at the significance level of a = 0.001.

• Yes, there is evidence at the significance level of a = 0.01 but not at a = 0.001.

• Yes, there is evidence at the significance level of a = 0.05 but not at a = 0.01.

• Yes, there is evidence at the significance level of a = 0.10 but not at a = 0.05.

• No, there is no evidence at the significance level of a = .10.

Meteorologists are interested in the relationship between minimum pressure and maximum wind speed of hurricanes. The minimum pressure, in millibars, and maximum wind speed, in knots, were collected for a random sample of 100 hurricanes from the year 1995 to the year 2012. A regression analysis of maximum wind speed on minimum wind pressure produced a 95 percent confidence interval of (-1.42, -1.20) for the slope of the least-squares regression line. Which statement is a correct interpretation of the interval?

• The probability is 0.95 that wind speed will decrease, on average, between 1.20 knots and 1.42 knots for each millibar increase in minimum pressure.

• The probability is 0.95 that a different sample of 100 hurricanes will result in an increase, on average, of wind speed between 1.20 knots and 1.42 knots for each millibar increase in minimum pressure.

• We can be 95% confident that wind speed decreases, on average, between 1.20 knots and 1.42 knots for each millibar increase in minimum pressure.

• We can be 95% confident that wind speed increases, on average, between 1.20 knots and 1.42 knots for each millibar increase in minimum pressure.

• We can be 95% confident that, for any sample of hurricanes, the wind speed will decrease, on average, between 1.20 knots and 1.42 knots for each millibar increase in minimum pressure.

What do we use to estimate unknown parameters?

• Hypothesis Test

• Confidence Interval

• Trig Identity

• Guess and check

Identify the confidence interval used in the figure below if each shaded area is equal to 0.05.

• 90%

• 95%

• 98%

• 99%

Percentiles of the bootstrap distribution are provided. Use the percentiles to report a 95% confidence interval for the parameter.

• 6.593 hours to 7.78 hours

• 6.322 hours to 8.082 hours

• 6.438 hours to 7.947 hours

• 6.174 hours to 8.304 hours

The manufacturer of a smart watch claims that individuals who pay attention to how many steps they take per day will inadvertently take more steps per day than individuals who pay no attention to how many steps they take per day. To investigate this claim, the manufacturer conducts a study to estimate the difference in the mean number of steps taken by those that pay attention to how many steps they take per day and those that do not. To do so, 40 volunteers are recruited. Half of the volunteers are randomly assigned to receive a smart watch and are taught how to use it to track their steps. The other half of the volunteers are given a wristband to wear, but are not informed that the wristband is tracking their steps. The volunteers are monitored for 30 days. The mean and standard deviation of the number of steps taken per day are computed for each group. The data are shown above. Which of the following is a 99% confidence interval for the difference in the mean number of steps taken by all people like these that do and do not pay attention to the number of steps they take per day using df = 19?

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To test a claim about a mean, when the population standard deviation is unknown we use:

• z procedures

• Pythagorean Theorem

• t procedures

• np > 10 and n(1-p) > 10

Do high school seniors with part-time jobs spend less time doing homework per week, on average, than seniors without part-time jobs? For a random sample of 45 seniors with part-time jobs, the mean amount of homework time is 4.2 hours with a standard deviation of 3.8 hours.  For a random sample of 45 seniors without part-time jobs, the mean amount of homework time is 5.8 hours with a standard deviation of 4.9 hours.  Assuming the conditions are met, which of the following is the correct standard error for a 95% confidence interval for a difference in population means?

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The makers of a specialty brand of bottled water claim that their “mini” bottles contain 8 ounces of water.  To investigate this claim, a consumer advocate randomly selected a sample of 10 bottles and carefully measured the amount of water in each bottle.  The mean volume was 7.98 ounces and the 95% confidence interval for the true mean volume is 7.93 to 8.03 ounces.  Based on the sample, which of the following conclusions best addresses the maker’s claim?

• Because 7.98 is in the interval, there is convincing evidence, that their claim is correct.

• Because 7.98 is in the interval, there is not convincing evidence, that their claim is incorrect.

• Because 8 is in the interval, there is convincing evidence, that their claim is correct.

• Because 8 is in the interval, there is not convincing evidence, that their claim is incorrect.

• Because 0 is in the interval, there is convincing evidence, that their claim is incorrect.