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Statistics : Standard Deviation and Mean (10 Problems)

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1. True or False?
The standard deviation of a population will always be smaller than the standard deviation of the sample means (of the samples used to estimate this same population).

2. The mean height for a population is 65 inches and the standard deviation is 3 inches. Let X_ denote the mean height for a sample of people picked randomly from the population. The standard
deviation of X_ for samples of size 30 is ________ the standard deviation of X_ for samples of size 20?
a. greater than
b. less than
c. equal to
d. impossible to tell
3. The error which results from using a sample to estimate a population characteristic is called the ___________.
a. rounding error
b. sampling error
c. statistical error
d. error interval

5. The mean height for a population is 65 inches and the standard deviation is 3 inches. Let A and B denote the events described below.
Event A: The mean height of a random sample of 16 people will be within 1 inch of the population mean.
Event B: The mean height of a random sample of 50 people will be within 1 inch of the population mean.
a. The probability of event A is greater than the probability of event B
b. The probability of event B is greater than the probability of event A
c. The probability of event A is equal to the probability of event B
d. The probability of event A is unrelated to the probability of event B

7. When using sample means to estimate a population mean, increasing the size of the sample will result in:
a. smaller sampling error and OX will decrease.
b. smaller sampling error and OX will increase.
c. greater sampling error and OX will decrease.
d.greater sampling error and OX will increase
8. The mean price of a new mobile home is $43800. The standard deviation of the price is $7200. Let X_ denote the mean price of a sample of new mobile homes. For samples of size 50, find the mean price.
Use only numbers in your answer. Do not use a dollar sign, do not use a comma, do not use a decimal point, and do not write any words, such as "dollars".
9. The mean price of a new mobile home is $43800. The standard deviation of the price is $7200. Let X_ denote the mean price of a sample of new mobile homes. For samples of size 50, find the standard deviation of
of X_. Round to the nearest cent (two decimal places). Use only numbers and a decimal point in your answer. Do not use a dollar sign, do not use a comma, and do not write any words, such as "dollars".
10. The table reports the distribution of pocket money, in bills, of the 6 students in a statistics seminar.
Student Bridgette Timothy Aaron Barry Lucille Stacey
Dollars 2 4 4 5 5 7
For a random sample of size two, find the probability that the sample mean will be within $1 of the population mean. Round to the nearest three decimal places.
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Ten problems involving Standard Deviation and Mean are solved.

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10 Problems on Descriptive Statistics, Point Estimates and Confidence Interval

1. Use the confidence interval to find the estimated margin error. Then find the sample mean.
A biologist reports a confidence interval of (1.8, 3.0) when estimating the mean height (in centimeters) of a sample of seedlings.

The estimated margin of error is______.

The sample mean is _________.

2. You work for a consumer advocate agency and want to find the mean repair cost of a washing machine. As part of your study, you randomly select 60 repair costs and find the mean to be $122.00. The sample standard deviation is $18.10. Complete parts (a) and (b).

(a) Construct a 95% confidence interval for the population mean repair cost.
The 95% confidence interval is (_____,________). ( Round to two decimal places as needed.)
(b) Change the sample size to n=120. Construct a 95% confidence interval for the population mean repair cost.
The 95% confidence interval is (______,______). (Round to two decimal places as needed.)

Which confidence interval is wider? Explain. Choose the correct answer below.

( ) The n=60 confidence interval is wider because a smaller sample is taken, giving less information about the population.
( ) The n=120 confidence interval is wider because a larger sample is taken, giving more information about the population.
( ) The two intervals are the same size because the confidence interval is based on the level of confidence and sample standard deviation.

3. A machine cuts plastic into sheets that are 30 feet (360 inches) long. Assume that the population of lengths is normally distributed. Complete parts (a) and (b).

(a) The company wants to estimate the mean length the machine is cutting the plastic within 0.25 inch. Determine the minimum sample size required to construct a 95% confidence interval for the population mean. Assume the population standard deviation is 0.5 inch.
n=_______. (Round up to the nearest whole number as needed.)

(b) Repeat part (a) using an error tolerance of 0.125 inch.
n= ________. (Round up to the nearest whole number as needed.)

Which error tolerance requires a larger sample size? Explain.

( ) The tolerance E=0.125 inch requires a larger sample size. As error size decreases, a larger sample must be taken to ensure the desired accuracy.
( ) The tolerance E= 0.125 inch requires a larger sample size. As error size increases, a larger sample must be taken to ensure the desired accuracy.
( ) The tolerance E=0.25 inch requires a larger sample size. As error size increases, a larger sample must be taken to ensure the desired accuracy.
( ) The tolerance E= 0.25 inch requires a larger sample size. As error decreases, a larger sample must be taken to ensure the desired accuracy.

4. The grade point averages (GPA) for 12 randomly selected college students are
2. 2 3.3 2.7
1.7 0.6 4.0
2.1 1.4 3.8
0.4 2.2 3.4

Assume the population is normally distributed.
(a) Find the sample mean.

Type equation here.
Sample mean=_________(Round to two decimal places as needed.)

(b) Find the sample deviation.
s=__________. (Round to two decimal places as needed.)

(c ) Construct a 90% confidence interval for the population mean.
A 90% confidence interval for the population mean is (______,______). (Round to two decimal places as needed.

5. In a random sample of 36 bolts, the mean length was 2.14 inches and the standard deviation was 0.08 inch. Use a normal distribution or a t-distribution to construct the 90% confidence interval for the mean.
Which distribution should be used to construct the 90% confidence interval?

( ) Use a normal distribution because the lengths are normally distributed and the standard deviation is known.
( ) Use a normal distribution because n > 30.
( ) Use a t-distribution because n > 30.
( ) Use a t-distribution because the lengths are normally distributed and the standard deviation is known.
The 905% confidence interval is (______,________). ( Round to two decimal places as needed.)

6. Let p be the population proportion for the following condition. Find the point estimates for p and q.

A study of 4436 adults from country A found that 2545 were obese or overweight.

The point estimate for p, ^p, is _______. (Round to three decimal places as needed.)

The point estimate for q, ^q, is _______. (Round to three decimal
Places as needed.)

7. The table below shows the results of a survey in which 400 adults from the East, 400 adults from the South, 400 adults from the Midwest, and 400 adults from the West were asked if traffic congestion is a serious problem. Complete parts (a) and (b).

Adults who say traffic
Congestion is a serious problem
East 37%
South 32 %
Midwest 25 %
West 57 %

(a) Construct a 99% confidence interval for proportion of adults from the Midwest who say traffic congestion is a serious problem.
The 99% confidence interval for the population of adults from the Midwest who say traffic congestion is a serious problem is (______,_______). ( Round to three decimal places as needed).

(b) Construct a 99% confidence interval for the proportion of adults from the West who say traffic congestion is a serious problem. Is it possible that these two proportions are equal? Explain your reasoning.

The 99% confidence interval for the proportion of adults from the West who say traffic congestion is a serious problem is (_______,________).
(Round to three decimal places as needed.)

Is it possible that these two proportions are equal?
( ) Yes, because 99% confidence interval for the Midwest overlaps with 99% confidence interval for the west.

( ) No, because the 99% confidence interval for the Midwest does not overlap with the 99% confidence interval for the west.

8. A researcher wishes to estimate, with 95% confidence, the percentage of adults who support abolishing the penny. His estimate must be accurate within 4% of the true proportion.

(a) Find the minimum sample size needed, using a prior study th(a)at found that 34% of respondents said they support abolishing the penny?

n=______(Round up to the nearest whole number as needed.)

(b) No preliminary estimate is available. Find the minimum sample size needed.

n= ______(Round up to the nearest whole number as needed.)

9. Find the critical values x²L and X²R for the given confidences level c and sample size n.
C=0.99 , n=20
X²L = __ (Round to three decimal places as needed.)
X²R- __ (Round to three decimal places as needed.)

10. You randomly select and measure the contents of 10 bottles of cough syrup. The results (in fluid ounces) are shown below
4.215 4.297 4.252 4. 244 4.189
4.274 4. 263 4.427 4.221 4.231
Assume the sample is taken from a normally distributed population. Construct 80% confidence intervals for (a) the population variance and (b) the population standard deviation.

(a) The confidence interval for the population variance is (______,_______). (Round to six decimal places as needed.)
(b) The confidence interval for the population standard deviation is (____,____). (Round to four decimal places as needed.).

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