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 sampling distribution is approximately normal due to a x̅= of 80 σx̅= 2.

  2. P (x̅> 83)= 0.0668. If you take 100 random sample with this same criteria then, about 7 of the samples will result in a mean that is less than or equal to 75.8.

  3. P (x̅ ≤ 75.8)=0.0179. If you take 100 random sample with this same criteria then, about 2 of the samples will result in a mean that is less than or equal to 75.8.

(d)P (78.3< x̅<85.1 )=0.7969 If you take 100 random sample with this same criteria then, about 80 of the samples will result in a mean that is between 78.3 and 85.1.

17

  1. The population has to be normally distributed if one is using the sample mean. When the population is normally distributed, so is the sampling distribution with a Ux̅=64 and σx̅= 4.907

  2. P (x̅ <67.3)=0.7486 . If you take 100 random sample with this same criteria then, about 75 of the samples will result in a mean that is less than 67.3.

  3. P (x̅ ≥65.2)= 0.4052. If you take 100 random sample with this same criteria then, about 41 samples will result in a mean that is greater than or equal to 65.2

19

  1. The probabilty that a randomly selected pregnacy lasts less than 260 days is 0.3520.

  2. The sampling distribution of the sample mean of the length of human pregnancies is normal with a Ux̅=266 and σx̅= 3.578.

  3. The probablity that a random sample of 20 pregancies has a mean gestation of 260 days or less is 0.0465. If you take 100 random sample with this same criteria then, about 5 samples will have mean gestation of 260 days or less.

  4. The probablity that a random sample of 50 pregancies has a mean gestation of 260 days or less is 0.0040. If you take 1000 random sample with this same criteria then, about 4 samples will have mean gestation of 260 days or less.

  5. If a random sample of 50 prgancies resulted in a mean gestaations periods of 260 days or less, then the sample must have come from a population that had a mean gestaations periods less than 266 days.

  6. The probablity that a random sample of 15 pregancies will have a mean gestation period within 10 days of the mean is 0.9844. If you take 100 random sample with this same criteria then, about 98 samples will have mean gestation period between 356 and 276 days.

21

  1. The probabilty that a randomly selected student will read more than 95 words per minute is 0.3085. If you take 100 random sample with this same criteria then,31 students will read more than 95 words per minute.

  2. The probabiltiy that a random sample of 12 second grade students results in a mean reading rate of more than 95 words per minute is 0.0418. If you take 100 random sample with this same criteria then, about 4 students will result in a mean reading rate of more than 95 words per minute.

  3. The probabiltiy that a random sample of 24 second grae students results in a mean reading rate of more than 95 words per minute is 0.0071. If you take 100 random sample with this same criteria then, about 7 students will result in a mean reading rate of more than 95 words per minute.

  4. Increasing the sample size decreases the probability of sample mean > 95.

  5. Mean reading of 92.8 wpm is unsual becaue p (sample mean greater than or equal to 92.8) = 0.1056. This means new reading program not abudantly more effective than old program.

  6. 5% chance that mean reading speed of random sample of 20 second-grade students will exceed 93.7 words

23

  1. The probability that a randomly selected month has a positive rate of return is 0.5675.

  2. With the next 12 months as a simple random sample, the probability that the mean monthly rate of return will be positive is 0.7291. If you take 100 random sample with this same criteria then, 73 samples will have a mean monthly rate of return be positive.

  3. With the next 24 months as a simple random sample, the probability that the mean monthly rate of return will be positive is 0.8051. 81 samples will have a mean monthly rate of return be positive.

  4. With the next 36 months as a simple random sample, the probability that the mean monthly rate of return will be positive is 0.8521. 85 samples will have a mean monthly rate of return be positive.

  5. The likelihood of earning a positive rate of return on stocks as the investment time horizon 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 sampling distribution is approximately normal with p-hat = 0.046

  2. The probability of obtaining x=63 or more individuals with the charactistic is 0.1922. About 19 out of 100 samples will result with 63 or more individuals and the characteristic.

  3. The probability of obtaining x=51 or fewer individuals with the charactistic is 0.0047. 63 or more individuals will result with the characteristic. About 5 out of 1000 samples will result with 51 or fewer individuals and the characteristic.

12

  1. The sampling distrbution is approximately normal with p-hat = 0.0337

  2. The probability of obtaining x=136 or more individuals with the charactistic is 0.1894. About 19 out of 100 samples will result with 136 or more individuals and the characteristic.

  3. The probability of obtaining x=118 or fewer individuals with the charactistic is 0.0392. About 39 out of 1000 samples will result with 118 or fewer individuals and the characteristic.

13

  1. The sampling distribution is approximately normal with the sampling distribution of p-hat = 0.015

  2. The probability of obtaining x=390 or more individuals with the charactistic is 0.0040. About 4 out of 1000 samples will result with 390 or more individuals and the characteristic.

  3. The probability of obtaining x=320 or fewer individuals with the charactistic is 0.0233. About 2 out of 100 samples will result with 320 or fewer individuals and the characteristi

14

  1. The sampling distribution of p̂ is Up= 0.42 and σp= 0.013.

  2. The probability of obtaining x=657 or more individuals with the charactistic is 0.0104. About 1 in a 100 samples will result with 657 or more individuals with the charactistic.

  3. The probability of obtaining x=584 or fewer individuals with the charactistic is 0.0618. About 6 in a 100 samples will result with 584 or fewer individuals with the charactistic. 15

  4. qualitative with 2 possible outcomes-order a mean in a foreign language, or not.

  5. The source of the variability is the individuals in the survey and their ability to order a meal in a foreign languaage

  6. p̂ is Up̅= 0.82 and σp̅= 0.035

  7. The probability the proportion who are satisfied with the way things are going in their life exceeds 0.85 is 0.2148. About 21 in 100 samples will result in people with the characteristic.

  8. If 75 or fewer are satistfied with the way things are going in their life probability is 0.0329. About 3 out of 100 samples will result with 75 or fewer individuals and the characteristic which would be unusual.

16

  1. Qualitative because you are referring to a personal opinion which relates to a qulaitative response.

  2. The source of variability is the Americans who are satisfied with the way things are going in their lives.

  3. The sampling distribution of p̂ is Up̅= 0.82 and σp̅= 0.038.

  4. The probability the proportion who are satisfied with the way things are going in their life exceeds 0.85 is 0.2148. About 21 in 100 samples will result in people with the characteristic.

  5. If 75 or fewer are satistfied with the way things are going in their life probability is 0.0329. About 3 out of 100 samples will result with 75 or fewer individuals and the characteristic which would be unusual.

17

  1. The sampling distribution of p̂ is Up̅=0.39 and σx̅= 0.022.

(b)The probability that in a random sample of 500 adult Americans less than 38% believe that marriage is obsolete is 0.3198. About 32 out of 100 samples will result in fewer than 190 individuals and the characteristic.

  1. The probability that in a random sample of 500 adult Americans between 40% and 45% believe that marriage is obsolete is 0.3198. About 32 out of 100 samples will result between 200 and 225 individuals and the characteristic.

  2. If 210 or more who believe marriage is obsolete, it would not be unusual. The probability is 0.0838 so about 8 out of 100 samples will result with 210 or more individuals and the characteristic.