To Be Confident or To Be Probable? That is the Question.
True or False: Confidence intervals are random events.
False
What is the critical value for a 95% confidence interval of a population mean when the population standard deviation is known?
1.96
Estimate the sample size needed to construct a 95% confidence interval for a population mean with a margin of error of 1.3 and a population standard deviation of 15.7.
561
Interpret the following 98% confidence interval for the population mean: (5.23, 7.65).
We are 98% confident that the true population mean lies between 5.23 and 7.65.
Construct a 98% confidence interval for a population proportion given a sample proportion of 0.543 in a sample of 200 individuals.
(0.461,0.625)
What is an assumption about samples when constructing ALL confidence intervals?
Simple Random Sample
True or False: A method used to construct confidence intervals is a random event.
True
What is the critical value for a 98% confidence interval of a population proportion?
2.326
Estimate the sample size needed to construct a 98% confidence interval for a population mean with a margin of error of 1.3 and a sample standard deviation of 15.7 from a previous sample of size 52.
790
Interpret the following 93% confidence interval for the population proportion: 0.452 +- 0.133.
We are 93% confident that the true population proportion lies between 0.319 and 0.585.
Construct a 90% confidence interval for a population proportion given 12 individuals of 63 sampled have a desired characteristic.
(0.109,0.272)
Name an assumption for constructing a confidence interval for a population mean when the standard deviation is unknown.
Sample is large (n > 30) or population is approximately normal.
A researcher claims that the probability is 0.99 that the confidence interval constructed using an appropriate method for a given sample of data will cover the population mean. Is this statement correct?
Yes, it is correct.
What is the critical value for a 95% confidence interval of a population mean when the population standard deviation is unknown and n=10?
2.262
Estimate the sample size needed to construct a 90% confidence interval for a population proportion with a margin of error of 0.6 and a sample proportion of 0.278 from a previous sample.
151
This confidence interval represents the proportion of voters in agreement with a certain legislation: (0.125,0.154)
A researcher had previously claimed 14% of the voters agree with the legislation. Does your confidence interval support or contradict this claim?
Because 0.14 is within the 95% confidence interval, the interval supports the researcher’s claim.
Construct a 99% confidence interval for a population proportion given a sample of 12 individuals and 3 with the desired quality.
(0.014,0.611)
Name an assumption for constructing a confidence interval for a population mean when the standard deviation is known.
Sample is large (n > 30) or population is approximately normal.
After calculating a confidence interval for the population mean, a researcher claims that the probability is 0.99 that the population mean is between 4.2 and 6.4. Is this statement correct?
No, this statement is incorrect.
Construct a 99% confidence interval for a population mean if xbar= 23.8, n=49 and sigma=0.93.
(23.46,24.14)
Estimate the sample size needed to construct a 90% confidence interval for a population proportion with a margin of error of 0.6 and no previous sample proportion available.
A government official suspects that the mean age of licensed drivers is getting older. He thinks that reports that the mean age of drivers is 38.4 is fraudulent. Therefore, he constructs a 98% confidence interval to estimate the mean age of drivers and finds it to be 36.8 +- 3.5. Did he uncover fraud?
The official did not uncover fraud. The confidence interval suggests that the true mean could indeed be 38.4.
Why might two confidence intervals constructed from samples of the same size and with the same confidence level yield different results?
Two confidence intervals from the same population, with the same sample size, and same confidence level can differ because each comes from a different sample. Therefore, each will have a different xbar or phat, resulting in different confidence intervals, even if the margins of error are the same.
This is why we say that we have a certain level of confidence that the interval will contain the true population mean or proportion or that the method of constructing the confidence interval has a probability equivalent to a certain level of confidence of covering the true population mean or proportion.
Name an assumption related to the population size for constructing a confidence interval for a population proportion.
The population is at least 20 times as large as the sample.
The population is divided into two categories.
Interpret the following 98% confidence interval: 0.263<p<0.428
Construct a 90% confidence interval for a population mean if xbar= 52.6, n=35 and s=1.3.
(52.23,52.97)
Does the margin of error, and thus the necessary sample size for a given margin of error, depend on the population size?
No, assuming a population is not very small.
Another government official suspects that the mean age of licensed drivers is getting younger. He thinks that reports that the mean age of drivers is 38.4 is fraudulent. Therefore, he constructs a 98% confidence interval to estimate the mean age of drivers and finds it to be 35.2 +- 1.3. Did he uncover fraud?
The confidence interval does not support the claim of 38.4. The official’s findings suggest the reported mean may be inaccurate.
Two government officials suspect that the mean age of licensed drivers is not accurate. Both think that reports that the mean age of drivers is 38.4 is fraudulent. Therefore, they both construct 98% confidence intervals to estimate the mean age of drivers. One finds it to be 36.8 +- 3.5. The other finds it to be 35.2 +- 1.3. Taken together, did they uncover fraud? Why or why not?
Taken together, the officials cannot definitively uncover fraud. One interval includes the reported mean, so there is still a reasonable possibility that the reported mean could be accurate.
Name an assumption related to the sample for constructing a confidence interval for a population proprtion.
The sample must contain at least 10 individuals in each category.