Are the requirements for constructing a confidence interval about p satisfied?

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You can use the following syntax to check your answers. Note the answers you get in R will not be EXACTLY the same you get by hand but they should be pretty close.

If your \(x = 542\) (ie the number of “successes” and your \(n = 3611\), this is how you can have R calculate a 90% Confidence Interval for you.

binom.test(x = 542, n = 3611,  conf.level = .90)[["conf.int"]]

## [1] 0.1403939 0.1602214
## attr(,"conf.level")
## [1] 0.9

25.

1. 0.150
2. The requirements for constructing a confidence interval about p are satisfied because the sample size is less than 5% of the population and (3611)(0.15)(0.85)>10.
3. Lower bound: 0.150. Upper bound: 0.160.
4. We are 90% confident that the population proportion of adult Americans who have used their smartphone to make a purchase is between 0.150 and 0.160.

26.

1. 0.430
2. The requirements for constructing a confidence interval about p are satisfied because the sample size is less than 5% of the population and (1153)(0.430)(0.570)>10.
3. Lower bound: 0.401. Upper bound: 0.458
4. We are 95% confident that the population proportion of workers and retirees in the United States 25 years of age and older who have less than $10,000 in savings is between 0.401 and 0.458.

27.

1. 0.519
2. The requirements for constructing a confidence interval about p are satisfied because the sample size is less than 5% of the population and (1003)(0.519)(0.481)>10.
3. Lower bound: 0.489. Upper bound: 0.551.
4. It’s possible that the true proportion isn’t within the interval and that a supermajority of adult Americans believe that television is a luxury they could do without. However, it’s unlikely because we can be 95% confident that population proportion of adult Americans who believe that televisions are a luxury is between 0.489 and 0.551.
5. Lower bound: 0.449. Upper bound: 0.511. We are 95% confident that the population proportion of adult Americans who believe that televisions are a necessity is between 0.449 and 0.511.

28.

1. 0.750
2. The requirements for constructing a confidence interval about p are satisfied because the sample size is less than 5% of the population and (1024)(0.750)(0.250)>10.
3. Lower bound: 0.715. Upper bound: 0.785.
4. It’s possible that the true proportion isn’t within the interval and that the proportion of adult Americans aged 18 or older for which the issue of family values is extremely or very important in determining their vote for president is below 70%. However, it’s unlikely because we can be 95% confident that the proportion is between 0.715 and 0.785.
5. Lower bound: 0.215. Upper bound: 0.585. We are 99% confident that the the proportion of adult Americans aged 18 or older for which the issue of family values is not extremely or very important in determining their vote for president is between 0.215 and 0.585.

29.

1. 0.540
2. The requirements for constructing a confidence interval about p are satisfied because the sample size is less than 5% of the population and (1748)(0.540)(0.460)>10.
3. Lower bound: 0.520. Upper bound: 0.560.
4. Lower bound: 0.509. Upper bound: 0.571.
5. Increasing the level of confidence widens the interval.

You can use the following syntax to check your answers. Note the answers you get in R will not be EXACTLY the same you get by hand but they should be pretty close.

If \(\bar{x} = 18.4\), sample standard deviation is 4.5, sample size = 35, and your confidence level is 95% this is how you can have R calculate a Confidence Interval for you.

conf_int(xbar = 18.4, size = 35, conf = .95, s = 4.5)

## [1] 16.8542 19.9458

21.

1. Lower bound: 16.85. Upper bound: 19.95.
2. Lower bound: 17.12. Upper bound: 19.68. The margin of error decreases when the sample size increases.
3. Lower bound: 16.32. Upper bound: 20.48. The margin of error increases when the level of confidence increases.
4. If the sample size is n = 15, the population must be normally distributed to compute the confidence interval.

23.

1. This interpretation is incorrect because it’s the interval that varies and not the mean.
2. This is a reasonable interpretation.
3. This is an unreasonable interpretation because it is using data about the mean to make assumptions about individuals.
4. This interpretation is incorrect because the it is using data about the mean number of hours worked by adult Americans to make assumptions about adults in Idaho.

25.

We are 90% confident that the mean drive-through time at Taco Bell is between 161.5 and 164.7 seconds.

27.

One could increase the sample size or decrease the level of confidence to decrease the confidence interval.

29.

1. The Central Limit Theorem states that the sampling distribution of the population mean becomes approximately normal as the sample size increases. Since the blood alcohol concentrations in fatal accidents is highly skewed right, a large enough sample size is needed for the sample mean distribution to be approximately normal.
2. This satisfies the requirements for constructing a confidence interval because the sample size is less than 5% of the population.
3. Lower bound: 0.1647. Upper bound: 0.1693. We are 90% confident that the mean BAC in fatal crashes in which the driver had a positive BAC is between 0.1647 and 0.1693 g/dL.
4. It’s possible that the true mean isn’t within the interval and that the mean is less than 0.08 g/dL but it’s unlikely.

31.

Lower bound: 12.05. Upper bound: 14.75. We can be 99% confident that the mean number of books that Americans read either all or part of during the preceding year is between 12.05 and 14.75.

33.

Lower bound: 1.08. Upper bound: 8.12. We can be 95% confident that the mean incubation period of the SARS virus is between 1.08 and 8.12 days.

You can use the following syntax to check your answers. Note the answers you get in R will not be EXACTLY the same you get by hand but they should be pretty close.

If you sample standard deviation s = 2, the sample size = 35, and your confidence level is 95% this is how you can have R calculate a Confidence Interval for \(\sigma\).

conf_sig(s = 2, size = 35, conf = .95)

## [1] 1.617744 2.620404

5

The critical values are 10.117 and 30.144.

7

The critical values are 9.542 and 40.289.

9

1. Lower bound: 7.94. Upper bound: 23.66.
2. Lower bound: 8.59. Upper bound: 20.63. Increasing the sample size decreases the width of the interval.
3. Lower bound: 6.61. Upper bound: 31.36. Increasing the level of confidince increases the width of the confidence interval.

11

Lower bound: 1.612. Upper bound: 4.278. We can be 95% confident that the population standard deviation of the prices of a 4 GB flash memory card is between 1.612 and 4.278 dollars.

13

Lower bound: 849.69. Upper bound: 1655.34. We can be 90% confident that the population standard deviation of the repair cost of a low-impact bumper cash on a mini- or micro-car is between 849.69 and 1655.34 dollars.

What are the requirements for constructing a confidence interval about p are satisfied?

What requirements must be satisfied in order to construct a confidence interval about a population proportion?

What are the requirements for constructing a confidence interval?

Are the requirements for constructing a confidence satisfied?