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 is basically normal.

  2. The probability is 0.0668

  3. Probability is 0.0179

  4. Probability is 0.7969

17

  1. The population needs to be normally distributed, and the sampling distribution is basically 4.097.

  2. The probability is 0.7486

  3. The probability is 0.4052

19

  1. The porbability is 0.03520

  2. The sampling distribution is normal and 3.578.

  3. The probability is 0.0456

  4. The probability is 0.004

  5. We would conclude that the sample came from a period that was likely less than 266 days.

  6. The probability is 0.9844.

21

  1. The probability is 0.3085.

  2. The probability is 0.0418.

  3. The probability is 0.0071.

  4. By increasing the sample size, the probability of the mean being greater than 95 is decreased. This is because the st dev decreases as the sample size increases.

  5. These findings don’t seem like they would be unusual- the new program also seems to not be effective.

  6. 93.7 words/minute.

23

  1. The probability is 0.5675.

  2. The probability is 0.7291.

  3. The probability is 0.8051

  4. The probability is 0.8531

  5. The likelihood of earning a positive rate of return increases.

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

8.2

11

  1. The sampling distribution is basically normal, with a standard deviation of 0.046 and a mean of 0.8.

  2. The probability is 0.1922

  3. The probability is 0.0047.

12

  1. The sampling distribution is approximately normal and 0.0337.

13

  1. The sampling distribution is approximately normal and 0.015.

  2. The probability is 0.0040.

  3. The probability is .0233.

14

  1. The distribution is approximately normal and 0.0129.

15

  1. It’s qualitative because it must be answered either yes or no.

  2. It’s a random variable because there is variance between the samples. The source of the variability comes from the individuals and their ability to order a meal in a foreign language.

  3. The proportion’s sampling distribution is basically normal and 0.035.

  4. The probability of this proportion is 0.1977.

  5. It would be unusual because based on the sample, the probability is 0.0239.

16

  1. It’s qualitative because it’s a yes or no answer.

  2. It’s a random variable because the samples are different- they vary in terms of data. The source of variability may be that the groups are a sample vs. the actual population.

  3. The sampling distribution is basically normal and 0.0384.

  5. It would be unusual because it is assumed from the figure of the population that the probability should be 82%, so that is lower than the assumed probability.

17

  1. The sampling distribution of the proportion is basically normal and 0.022.

  2. The probability is 0.3228.

  3. The probability is 0.3198.

  4. It’s not unsual because the probability from this situation turns out to be 0.083.