AP Stats - Confidence Intervals

A 95% confidence interval for the mean $\mu$  of a population is computed from a random sample and found to be  $9\pm3$  .  We may conclude that

• in a given random sample, there is a 95% probability that $\mu$ is between 6 and 12

• 95% of values sampled are between 6 and 12

• if we took many, many additional random samples and, from each, computed a 95% confidence interval for $\mu$ , approximately 95% of these intervals would contain $\mu$ .

• there is a 95% probability that the true mean is 9 and a 95%chance that the true margin of error is 3

• all of the above are true

The records of all 100 postal employees at a postal station in a large city showed that the average amount of time these employees had worked for the US Postal Service was $\overline{x}$  = 8 years.  Assume the distribution of the amount of time US Postal Service employees have spent with the Postal Service is approximately normal with standard deviation $\sigma=5\ years$  .  Based on these data, a 95% confidence interval for the mean number of years $\mu$  that a US Postal Service employee has spent with the Postal Service would be

•  $8\pm0.82$  

•  $8\pm0.98$  

•  $8\pm1.96$  

•  $8\pm9.80$  

•  $8\pm0.098$  

The scores of a certain population on the Wechscler Intelligence Scale for Children (WISC) are thought to be normally distributed with mean $\mu$  and standard deviation $\sigma=10$  .  A simple random sample of 10 children from this population is taken, and each is given the WISC.  The 95% confidence interval for $\mu$  ,  $\overline{x}\pm1.96\frac{10}{\sqrt{10}}$ , is computed from these scores.  A histogram of the 10 WISC scores is shown.

Based on this histogram, we would conclude that:

• the 95% confidence interval computed from these data is very reliable.

• the 95% confidence interval computed from these data is not very reliable.

• the 95% confidence interval computed from these data is actually a 99% confidence interval.

• the 95% confidence interval computed from these data is actually a 90% confidence interval

• only one student's score should fall outside the 95% confidence interval.

A 90% confidence interval for the mean $\mu$ of a population is computed from a random sample and is found to be $9\pm3.$    Which of the following could be the 95% confidence interval based on the same data?

•  $9\pm1.96$  

•  $9\pm2$  

•  $9\pm3$  

•  $9\pm4$  

• Without knowing the sample size, any of the above answers could be the 95% confidence interval

An agricultural researcher plants 25 plots with a new variety of corn. The average yield for those plots is $\overline{x}=150$ bushels per acre.  Assume that the yield per acre for the new variety of corn follows a normal distribution with unknown mean $\mu$  and standard deviation $\sigma=10$  bushels.  A 90% confidence interval for $\mu$  is:

•  $150\pm2.00$  

•  $150\pm3.29$  

•  $150\pm3.92$  

•  $150\pm16.45$  

•  $150\pm32.90$  

An agricultural researcher plants 25 plots with a new variety of corn. A 90% confidence interval for the average yield for those plots is found to be $162.72\pm4.47$ bushels per acre.  Which of the following would produce a confidence interval with a smaller margin of error than this 90% confidence interval?

• Choosing a sample with a larger standard deviation

• Planting 100 plots, rather than 25

• Choosing a sample with a smaller standard deviation

• Planting only 5 plots rather than 25

• None of the above

You measure the heights of a random sample of 400 high school sophomore males in a Midwestern state. You find the mean of these students to be $\overline{x}=66.2$ inches.  Suppose that the heights of all high school sophomore males follow a normal distribution with unknown mean $\mu$  and standard deviation $\sigma=4.1$ inches.  A 95% confidence interval for $\mu$  is: 

• (58.16,74.24)

• (58.46,72.94)

• (65.8, 66.6)

• (65.86, 66.54

• (66.18, 66.22)

You measure the heights of a random sample of 400 high school sophomore males in a Midwestern state. You find the mean of these students to be $\overline{x}=66.2$ inches.  Suppose that the heights of all high school sophomore males follow a normal distribution with unknown mean $\mu$ and standard deviation $\sigma=4.1$ inches.

I compute a 95% confidence interval for $\mu$ .  Suppose I has measured the heights of a random sample of 100 sophomore males, rather than 400.  Which of the following statements is true?

• The margin of error for our 95% confidence interval would increase.

• The margin of error for our 95% confidence interval would decrease.

• The margin of error for our 95% confidence interval would stay the same, since the level of confidence has not changed.

•  $\sigma$ would increase

•  $\sigma$ would decrease

Suppose that the population of the scores of all high school seniors who took the Math SAT (SAT-M) test this year follows a normal distribution with mean $\mu$ and standard deviation $\sigma=100.$  You read a report that says, "On the basis of a simple random sample of 100 high school seniors that took the SAT-M this year, a confidence interval for $\mu$ is $512\pm25.76."$   The confidence level for this interval is 

• 90%

• 95%

• 96%

• 99%

• over 99.9%

An agricultural researcher plants 25 plots with a new variety of corn. The average yield for these plots is $\overline{x}=150$ bushels per acre.  Assume that the yield per acre for the new variety of corn is normally distributed with unknown mean $\mu$ and that a 90% confidence interval for $\mu$ is found to be $150\pm3.29.$  What can we deduce about the standard deviation $\sigma$ of the yield per acre for the new variety of corn?  

•  $\sigma$  is about 10

•  $\sigma$ will be larger than 10 if we use 100 plots 

•  $\sigma$ is 3.29 

• If we repeated the experiment many, many additional times and from each computed a 90% confidence interval, $\sigma$ would be within 3.29 of the mean in approximately 90% of the intervals 

•  $\sigma$ will vary depending on the sample size