Statistics for Business Assignment
Answer all parts of the five questions below. For questions 2-5 please show all of your work.
Question 1 -
1) In its standardised form, the normal distribution
A) has an area equal to 0.5.
B) has a mean of 1 and a variance of 0.
C) has a mean of 0 and a standard deviation of 1.
D) Cannot be used to approximate discrete probability distributions.
2) Which of the following about the normal distribution is NOTtrue?
A) About 2/3 of the observations fall within ± 1 standard deviation from the mean.
B) It is a discrete probability distribution.
C) Its parameters are the mean, μ, and standard deviation, σ.
D) Theoretically, the mean, median, and mode are the same.
3) If a particular batch of data is approximately normally distributed, we would find that approximately
A) 19 of every 20 observations would fall between ± 2 standard deviations around the mean.
B) 4 of every 5 observations would fall between ± 1.28 standard deviations around the mean.
C) 2 of every 3 observations would fall between ± 1 standard deviation around the mean.
D) All the above.
4) The value of the cumulative standardised normal distribution at Z is 0.8770. The value of Z is
A) 0.18.
B) 1.16.
C) 0.81.
D) 1.47.
5) The owner of a fish market determined that the average weight for a rainbow trout is 3.2 kilograms with a standard deviation of 0.8 kilogram. A citation rainbow trout should be one of the top 2% in weight. Assuming the weights of rainbow trout are normally distributed, at what weight (in kilograms) should the citation designation be established?
A) 4.84 kilograms.
B) 5.20 kilograms.
C) 7.36 kilograms.
D) 1.56 kilograms.
6) The standard error of the mean
A) decreases as the sample size increases.
B) is never larger than the standard deviation of the population.
C) measures the variability of the mean from sample to sample.
D) All of the above.
7) The Central Limit Theorem is important in statistics because
A) for any sized sample, it says the sampling distribution of the sample mean is approximately normal.
B) for any population, it says the sampling distribution of the sample mean is approximately normal, regardless of the sample size.
C) for a large n, it says the population is approximately normal.
D) for a large n, it says the sampling distribution of the sample mean is approximately normal, regardless of the shape of the population.
8) Which of the following statements about the sampling distribution of the sample mean is incorrect?
A) The standard deviation of the sampling distribution of the sample mean is equal to σ.
B) The sampling distribution of the sample mean is generated by repeatedly taking samples of size n and computing the sample means.
C) The mean of the sampling distribution of the sample mean is equal to μ.
D) The sampling distribution of the sample mean is approximately normal whenever the sample size is sufficiently large (n ≥ 30).
9) Australian football salaries averaged $150,000 with a standard deviation of $80,000 in a recent year. Suppose a sample of 100 football players was taken. Find the approximate probability that the average salary of the 100 players exceeded $100,000.
A) Approximately 1
B) 0.2357
C) 0.7357
D) Approximately 0
10) For sample size 16, the sampling distribution of the mean will be approximately normally distributed
A) if the sample is normally distributed.
B) if the shape of the population is symmetrical.
C) if the sample standard deviation is known.
D) regardless of the shape of the population.
11) The standard error of the mean for a sample of 100 is 30. What is the standard deviation?
A) 400
B) 25
C) 300
D) 200
12) In the construction of confidence intervals, if all other quantities are unchanged, an increase in the sample size will lead to a _________ interval.
A) less significant
B) narrower
C) biased
D) wider
13) Suppose a 95% confidence interval for μ turns out to be (1,000, 2,100). Give a definition of what it means to be "95% confident" in an inference.
A) 95% of the observations in the entire population fall in the given interval.
B) In repeated sampling, the population parameter would fall in the given interval 95% of the time.
C) 95% of the observations in the sample fall in the given interval.
D) In repeated sampling, 95% of the intervals constructed would contain the population mean.
14) Private colleges rely on money contributed by individuals for much of their major building expenses. Much of this money is put into a fund called a trust, and the college spends only the interest earned by the fund. A recent survey of 8 private colleges in Australia revealed the following trusts (in millions of dollars): 60.2, 47.0, 235.1, 490.0, 122.6, 177.5, 95.4, and 220.0. What value will be used as the point estimate for the mean trust of all private colleges in Australia?
A) $180.975
B) $8
C) $143.042
D) $1,447.8
15) An economist is interested in studying the incomes of consumers in a particular region. The population standard deviation is known to be $2,000. A random sample of 50 individuals resulted in an average income of $30,000. What is the upper end point in a 99% confidence interval for the average income?
A) $30,282
B) $30,1042
C) $30,728
D) $30,660
16) Which of the following would be an appropriate null hypothesis?
A) The population proportion is less than 0.65.
B) The sample proportion is no less than 0.65.
C) The sample proportion is less than 0.65.
D) The population proportion is no less than 0.65.
17) Which of the following would be an appropriate alternative hypothesis?
A) The mean of a sample is greater than 55.
B) The mean of a population is greater than 55.
C) The mean of a sample is equal to 55.
D) The mean of a population is equal to 55.
18) A Type I error is committed when
A) we reject a null hypothesis that is false.
B) we don't reject a null hypothesis that is true.
C) we don't reject a null hypothesis that is false.
D) we reject a null hypothesis that is true.
19) If a test of hypothesis has a Type I error probability (α) of 0.01, it means that
A) if the null hypothesis is false, you reject it 1% of the time.
B) if the null hypothesis is true, you reject it 1% of the time.
C) if the null hypothesis is false, you don't reject it 1% of the time.
D) if the null hypothesis is true, you don't reject it 1% of the time.
20) The power of a statistical test is
A) the probability of not rejecting H0 when it is false.
B) the probability of rejecting H0 when it is false.
C) the probability of not rejecting H0 when it is true.
D) the probability of rejecting H0 when it is true.
Question 2 - Please show all your work.
The number of column centimetres of classified advertisements appearing on Mondays in a certain daily newspaper is normally distributed with population mean 320 and population standard deviation 20 cm.
With this information, answer the following questions:
1. For a randomly chosen Monday, what is the probability there will be less than 340 column centimetres of classified advertisement?
2. For a randomly chosen Monday, what is the probability there will be between 280 and 360 column centimetres of classified advertisement?
3. For a randomly chosen Monday the probability is 0.1 that there will be less than how many column centimetres of classified advertisements?
Question 3 - Please show all your work.
The mean selling price of new homes in a country town over a year was $115,000. The population standard deviation was $25,000. A random sample of 100 new home sales from this town was taken.
With this information, answer the following questions:
1. What is the probability that the sample mean selling price was more than $110,000?
2. What is the probability that the sample mean selling price was between $113,000 and $117,000?
3. What is the probability that the sample mean selling price was between $114,000 and $116,000?
Question 4 - Please show all your work.
To become an actuary, it is necessary to pass a series of 10 exams, including the most important one, an exam in probability and statistics. An insurance company wants to estimate the mean score on this exam for actuarial students who have enrolled in a special study program. They take a sample of 8 actuarial students in this program and determine that their scores are: 2, 5, 8, 8, 7, 6, 5, and 7. This sample will be used to calculate a 90% confidence interval for the mean score for actuarial students in the special study program.
1. What do you have to assume about the distribution of student test scores for the confidence interval to valid?
2. Please compute the degrees of freedom the confidence interval will be based on.
3. Please find the critical value used in constructing a 90% confidence interval for the student test scores.
4. Please compute the 90% confidence interval for the mean score of actuarial students in the special program.
Question 5 - Please show all your work.
A drug company is considering marketing a new local anaesthetic. The effective time of the anaesthetic the drug company is currently producing has a normal distribution with a mean of 7.4 minutes with a standard deviation of 1.2 minutes. The chemistry of the new anaesthetic is such that the effective time should be normal with the same standard deviation, but the mean effective time may be lower. If it is lower, the drug company will market the new anaesthetic; otherwise, it will continue to produce the older drug. A sample of size 36 results in a sample mean of 7.1. A hypothesis test will be done to help make the decision.
With this information, answer the following questions:
1. Please find the critical value for a test with a level of significance of 0.10.
2. Please compute the value of the test statistic.
3. Please find the p-value of the test.
4. Please determine the power of the test if the mean effective time of the anaesthetic is 7.0 using a 0.05 level of significance.
5. Please determine the probability of making a Type II error if the mean effective time of the anaesthetic is 7.0 using a 0.05 level of significance.