# final exam winter

A 95% confidence interval for p, the proportion of all shoppers at a large grocery store who purchase cookies, was found to be (0.236, 0.282). The point estimate and margin or error for this interval are:

(a) Point estimate = 0.236; Margin of error = 0.282
(b) Point estimate = 0.236; Margin or error = 0.046
(c) Point estimate unknown; Margin of error = 0.023
(d) Point estimate = 0.259; Margin or error = 0.046
(e) Point estimate = 0.259; Margin of error = 0.023

(e)Point estimate = 0.259; Margin error = 0.023
.236-.282=.046
.046/2=.023

.023+.236=.259

Which of the following is a correct statement?

(a) About 95% of the shoppers have between a 23.6% and a 28.2% chance of purchasing cookies.
(b) There is a 95% probability that the sample proportion lies between 0.236 and 0.282.
(c) If a second sample was taken, there is a 95% chance that its confidence interval would contain 0.25.
(d) This confidence interval indicates that more than 25% of shoppers buy cookies.
(e) We are reasonably certain the true proportion of shoppers who purchase cookies is between 24% and 28%.

(e)We are reasonably certain the true proportion of shoppers who purchase cookies is between 24% and 28%
You want to estimate the mean SAT score for a population of students with a 90% confidence interval. Assume that the population standard deviation is V = 100. If you want the margin of error to be no more than 10, you will need a minimum sample size of approximately

(a) 17
(b) 38
(c) 271
(d) 385
(e)1646

(c)271

(1-.9)/2=.05

1.64 (100/√ n) <= 10 (100/√ n)<=6.098 (100/6.098)^2=271

The government claims that students earn an average of \$4500 during their summer break from studies. A random sample of students gave a sample average of \$3975, and a 95% confidence interval was found to be \$3525 < u < \$4425. Which of the following is a correct interpretation of 95% confidence? (a) if the study were to be repeated many times, there is a 95% probability that the true average summer earnings is not \$4500 as the government claims. (b) because our specific confidence interval does not contain the value \$4500 there is a 95% probability that the true average summer earnings is not \$4500. (c) if we were to repeat our survey many times, then about 95% of all the confidence intervals will contain the value \$4500. (d) if we repeat our survey many times, then about 95% of our confidence intervals will contain the true value of the average earnings of students. (e) there is a 95% probability that the true average earnings are between \$3525 and \$4425 for all students
(d) if we repeat our survey many times, then about 95% of our confidence intervals will contain the true value of the average earnings of students.
An SRS of 50 students at a large middle school are asked, “Do you have a television in your bedroom? Twenty-eight of the students responded “yes.” If ˆ p = the proportion of students who answered “yes,” what is the standard error of ˆp ?

(a) 0.005
(b) 0.010
(c) 0.070
(d) 0.106
(e) The standard error cannot be calculated unless we know the standard deviation.

(c).070

√((.56 x .44)/50) = .0702

The t-confidence interval for a population mean is robust with respect to which of the following concerns?

(a)Under coverage
(b)Lack of randomness in sampling
(c)Skew in the population distribution
(d)the presence of extreme outliers
(e)None of the above.

(c)Skew in the population distribution
which of the following would be true about a 98% confidence interval constructed using the same data?

(a) The interval would be wider, because the standard error would be larger.
(b) The interval would be narrower, because the standard error would be smaller.
(c) The interval would be wider, because the critical z* would be larger.
(d) The interval would be narrower, because the critical z* would smaller.
(e) The interval would be about the same width, because the standard error would be smaller, but the critical z* would be larger. c

(c)The interval would be wider, because the critical z* would be smaller

Keywords: wider, critical z*, larger

Suppose that the population of the scores of all high school seniors who took the SAT Math test
this year follows a normal distribution with mean ju and standard deviation a = 100. You read a
report that says, “on the basis of a simple random sample of 100 high school seniors that took the
SAT-M test this year, a confidence interval for ja is 512.00 ± 25.76.” The confidence level for this
interval is

(a)90%
(b)95%
(c)99%
(d)99.5%
(e)over 99.9%

(c) 99%

25.76=z* (100/√100)
25.76/10=z*
z*=2.576
…99%

In a study of the effects of acid rain, a random sample of 100 trees from a particular forest is examined.
Forty percent of the trees show some signs of damage. Which of the following statements is correct?

(a) 40% is a parameter
(b) 40% is a statistic
(c). 40% of all trees in the forest show some signs of damage
(d). More than 40% of the trees in the forest show some signs of damage
(e). Less than 40% of the trees in the forest show some signs of damage

(b)40% is a statistic

keywords: sample, statistic

The sampling distribution of a statistic is

(a) the probability that we obtain the statistic in repeated random samples.
(b) the mechanism that determines whether randomization was effective.
(c) the distribution of values taken by a statistic in all possible samples of the same sample size from the same population.
(d) the extent to which the sample results differ systematically from the truth.
(e) the distribution of values in a sample of size n from the population

(c)the distribution of values taken by a statistic in all possible samples of the same sample size from the same population.

keywords: all possible samples, same sample size

A statistic is said to be unbiased if

(a) the survey used to obtain the statistic was designed so as to avoid even the hint of racial or sexual prejudice.
(b) the mean of its sampling distribution is equal to the true value of the parameter being estimated.
(c)both the person who calculated the statistic and the subjects whose responses make up the statistic were truthful.
(d)the value from any sample is equal to the parameter being estimated.
(e)it is used for honest purposes only.

(c)the mean of its sampling distribution is equal to the true vale of the parameter being estimated.

keywords: true mean equal

Which of the following distributions has a mean that varies from sample to sample?
I. The population distribution
II. The distribution of sample data
III. The sampling distribution

(a) I only
(b) II only
(c) III only
(d) II and III
(e) all three distributions

(b)II only
The distribution of sample data
X and Y are independent random variables, and a and b are constants. Which one of the following
statements is true?

(a)σX + Y = σX+σY
(b)Var (X – Y) = Var (X) + Var (Y)
(c)Var (a + bX) = b Var (X)
(d)σX – Y = σX – σY
(e)Var (X + Y) = Var (X) + Var (Y)

(b)Var(X-Y)=Var(x)+Var(y)
Let the random variable X represent the profit made on a randomly selected day by a certain store. Assume that X is Normal with mean \$360 and standard deviation \$50. What is the value of P(X > \$400)?

A) 0.2119 B) 0.2881 C) 0.7881 D) 0.845
E) 0.9999

(a).2119
Which of the following are true statements?

(a) The expected value of a geometric random variable is determined by the formula p(1-p)^n-1
(b) The distribution of every binomial random variable is skewed right
(c) If X is a geometric random variable and the
probability of success is .85, then the probability distribution of X will be skewed left, snince 85 is closer to 1 than to 0.
(d) An important difference between binomial and geometric random variables is that there is a fixed number of trials in a binomial setting, and the number of trials varies in a geometric setting.

(c)
A marketing survey compiled data on the number of personal computers in households. If X = the
number of computers in a randomly-selected household, and we omit the rare cases of more than 5 computers, then X has the following distribution:

X 0 1 2 3 4 5
P(X) 0.24 0.37 0.20 0.11 0.05 0.03

What is the probability that a randomly chosen household has at least two personal computers?

A. 0.19
B. 0.20
C. 0.29
D. 0.39
E. 0.61

(d)0.39
P(X>= 2)=1
P(x=0 or x=1) =1
(.24+.37)-1=.39
A random variable X has a probability distribution as follows:
X 0 1 2
3
P(X) 2k 3k 13k 2k
Where k is a positive constant. The probability P(X < 2.0) is equal to A. 0.90. B. 0.25. C. 0.65. D. 0.15. E. 1.00.
(b)0.25

2k+3k+13k+2k=1
k=.05
p(X<2)=P(x=0 or x=1)=5k=.25

Roll a fair 8-sided die 10 times. The probability of getting exactly 3 sevens in those 10 rolls is given by
(b)(10) (1)^3 (7)^5
(3) (8) (8)
(b)
Binomial probability formula: P(k successes in n trials when p = success in one trial) is
binomial setting
consists of n independent trials of the same chance process, each resulting in a success or a failure, with probability of success p on each trial. The count X of successes is a binomial random variable. Its probability distribution is a binomial distribution.