Here is some code that will allow you to check your answers. For example if \(\bar{X} \sim \mathcal{N}(5,2)\) and you want to know what the \(Pr(\bar{X} < 3)\) is, you can use the following code. Meaning just change the mean and sd to fit the problem you are #working on.

pnorm(2, mean = 5, sd = 2)
## [1] 0.0668072

8.1

15

  1. The mean of x bar is 80 and the standard deviation is 2.

  2. 0.0668

  3. 0.0179

  4. 0.7969

17

  1. In order to use the normal model to compute probabilities involving the sample mean, the distribution of the population must be normally distributed. The mean of x bar is 64 and the standard deviation is 4.907.

  2. 0.7486

  3. 0.4052

19

  1. 0.3520

  2. The mean of x bar is 266 days and the standard deviation is 3.578.

  3. 0.0465

  4. 0.0040

  5. If the mean gestation period of 50 pregnancies is 260 days or less, then the mean of this specific population must be lower than the average of 266 days.

  6. 0.9844

21

  1. 0.3085

  2. 0.0418

  3. 0.0071

  4. Increasing the sample size decreases the probility because as the sample size increases, the standard deviation decreases.

  5. I can conclude that the teacher’s new reading program is not extremely effecctive since the the probility that 20 students will have a mean above 92.8 is 0.1056, so a mean reading rate of 92.8wpm for 20 students is not unusual.

  6. 93.7 wpm

23

  1. 0.5675

  2. 0.7291

  3. 0.8051

  4. 0.8531

  5. The likelyhood of earning a positive rate of return on stocks increases as the investment time horizon increases.

Similarily you can use the above code to determine the \(Pr(\hat{P} < \hat{p})\)

8.2

11

  1. The mean of p hat is 0.8 and the standard deviation is 0.046.

  2. 0.1922

  3. 0.0047

12

  1. The mean of p hat is 0.65 and the standard deviation is 0.032.

  2. 0.1762

  3. 0.0025

13

  1. The mean of p hat is 0.35 and the standard deviation is 0.015.

  2. 0.0040

  3. 0.0233

14

  1. The mean of p hat is 0.42 and the standard deviation is .013.

  2. 0.0107

  3. 0.0643

15

  1. Qualitative because the answer is not numerical and there are two categorical options.

  2. The sample proportion is a random variable because the source of the variability is the random individuals answering the survey quesetion.

  3. The mean of p hat is 0.47 and the standard deviation is 0.035.

  4. 0.1977

  5. This result would be unusual because the probability that 80 out if 200 Americans can order a meal in a foreign language is only 0.0239.

16

  1. Qualitative because the answer is not numerical and there are two categorical options, yes or no.

  2. The sample proportion is a random variable because the source of the variability is the random individuals answering the survey quesetion.

  3. The mean of p hat is 0.82 and the standard deviation is 0.038.

  4. 0.2177

  5. This result would be unusual because the probability that 75 out if 100 Americans are satisfied with their lives is only 0.0329.

17

  1. The mean of p hat is 0.39 and the standard deviation is 0.022.

  2. 0.3228

  3. 0.3198

  4. This result would not be unusual because the probability that 210 out of 500 Americans believe marraige is obsolete is 0.0838.