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. It’s approximately normally distributed and the sample mean = 80 and the sample standard deviation = 2

  2. P(x > 83) = .0668

  3. = .0179

  4. = .7969

17

  1. It must be be normally distributed to be able to use the normal model. If thats true, then the sample mean = 64 and the sample st dev = 4.91

  2. = .7486

  3. = .4052

19

  1. = .3557

  2. It’s normally distributed, the sample mean = 266 days and the sample st dev = 3.5778

  3. .0485

  4. .0040

  5. This would be an extremely abnormal pregnancy, so

  6. = .9844

21

  1. .3085

  2. .0418

  3. .0071

  4. Larger sample sizes lead to less of a probability because the standard deviation is getting smaller as the population is getting larger.

  5. The probability is .1056 so the mean of 92.8 is not extremely unusual.

  6. The probability that the mean of a sample of 20 second grade students exceeds 93.7 wpm is .05

23

  1. .5675

  2. .7257

  3. .8023

  4. .8508

  5. the possibility of getting a positive rate of return increases as the time decreases.

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

8.2

11

  1. The distribution is normal and the sample probability mean is = .8 whereas the sample probability st dev is = .04618

  2. .1949

  3. .0048

12

  1. The distribution is approximately normal and the sample probability mean is = .65 whereas the sample probability st dev is = .03372

  2. .1894

  3. .0384

13

  1. approx norm dist and sample probability mean is = .35 whereas the sample probability st dev is = .015

  2. .0042

  3. .0233

14

  1. approx norm dist and sample probability mean is = .42 whereas the sample probability st dev is = .0129

  2. .0102

  3. .0606

15

  1. Qualitative because the response is whether they can order the meal in a foreign language

  2. The variability comes from the randomly selected individuals and whether or not they can order a meal in a foreign language

  3. approx norm dist and sample probability mean is = .47 whereas the sample probability st dev is = .03529

  4. .1977

  5. .0239 This is unusual because only 2percent of the samples will have this result

16

  1. Qualitative because it’s a yes or no response

  2. There is variability from the random selection and the different individuals

  3. approx norm dist and sample probability mean is = .82 whereas the sample probability st dev is = .03841

  4. .2177

  5. .0344 yes this is unusual because theres only a 3 percent chance that it would occur

17

  1. approx norm dist and sample probability mean is = .39 whereas the sample probability st dev is = .02181

  2. .3264

  3. .3234

  4. .0853 This is not highly unusual because about 8 of the 100 individuals would fall under this characteristic.