In: Math

* You wanted to estimate the mean number of vehicles crossing a busy bridge in your neighborhood each morning during rush hour for the past year. To accomplish this, you stationed yourself and a few assistants at one end of the bridge on 31 randomly selected mornings during the year and counted the number of vehicles crossing the bridge in a 10-minute period during rush hour. You found the mean to be 119 vehicles per minute, with a standard deviation of 31.

(a**) Construct the 95% confidence interval for the
population mean (vehicles per minute).**

**.................. -
.......................**

(b) **Construct the 99% confidence interval for the
population mean (vehicles per minute)**

..................... - .....................

* A college counselor wants to determine the average amount of time first-year students spend studying. He randomly samples 31 students from the freshman class and asks them how many hours a week they study. The mean of the resulting scores is 17 hours, and the standard deviation is 5.8 hours.

**(a) Construct the 95% confidence interval for the
population mean.**

................— ...............

**(b) Construct the 99% confidence interval for the
population mean.**

....................... - ....................

*A cognitive psychologist believes that a particular drug improves short-term memory. The drug is safe, with no side effects. An experiment is conducted in which 8 randomly selected subjects are given the drug and then given a short time to memorize a list of 10 words. The subjects are then tested for retention 15 minutes after the memorization period. The number of words correctly recalled by each subject is as follows: 6, 11, 12, 4, 6, 7, 8, 6. Over the past few years, the psychologist has collected a lot of data using this task with similar subjects. Although he has lost the original data, he remembers that the mean was 4.9 words correctly recalled and that the data were normally distributed.

(a) **On the basis of these data, what can we conclude
about the effect of the drug on short-term memory? Use α =
0.05 _{2 tail} in making your decision.**

t_{obt} = |

*t*_{crit} = ±

Question 1

Confidence Interval

X̅ ± t(α/2, n-1) S/√(n)

t(α/2, n-1) = t(0.05 /2, 31- 1 ) = 2.042

119 ± t(0.05/2, 31 -1) * 31/√(31)

Lower Limit = 119 - t(0.05/2, 31 -1) 31/√(31)

Lower Limit = 107.6306

Upper Limit = 119 + t(0.05/2, 31 -1) 31/√(31)

Upper Limit = 130.3694

**95% Confidence interval is ( 107.6306 , 130.3694
)**

Confidence Interval

X̅ ± t(α/2, n-1) S/√(n)

t(α/2, n-1) = t(0.01 /2, 31- 1 ) = 2.75

119 ± t(0.01/2, 31 -1) * 31/√(31)

Lower Limit = 119 - t(0.01/2, 31 -1) 31/√(31)

Lower Limit = 103.6886

Upper Limit = 119 + t(0.01/2, 31 -1) 31/√(31)

Upper Limit = 134.3114

**99% Confidence interval is ( 103.6886 , 134.3114
)**

Question 2

Confidence Interval

X̅ ± t(α/2, n-1) S/√(n)

t(α/2, n-1) = t(0.05 /2, 31- 1 ) = 2.042

17 ± t(0.05/2, 31 -1) * 5.8/√(31)

Lower Limit = 17 - t(0.05/2, 31 -1) 5.8/√(31)

Lower Limit = 14.8728

Upper Limit = 17 + t(0.05/2, 31 -1) 5.8/√(31)

Upper Limit = 19.1272

**95% Confidence interval is ( 14.8728 , 19.1272
)**

Confidence Interval

X̅ ± t(α/2, n-1) S/√(n)

t(α/2, n-1) = t(0.01 /2, 31- 1 ) = 2.75

17 ± t(0.01/2, 31 -1) * 5.8/√(31)

Lower Limit = 17 - t(0.01/2, 31 -1) 5.8/√(31)

Lower Limit = 14.1353

Upper Limit = 17 + t(0.01/2, 31 -1) 5.8/√(31)

Upper Limit = 19.8647

**99% Confidence interval is ( 14.1353 , 19.8647
)**

A company that produces 8-oz low-fat yogurt cups wanted to
estimate the mean number of calories for such cups. A random sample
of 10 such cups produced the following numbers of calories: Assume
that the number of calories in such cups are normally
distributed.
147 159 153 146 144 148 163 153 143 158
a. What is the point estimate of the mean calories for 8-oz low-fat
yogurt cups?
b. Construct a 95% confidence interval for the mean calories for...

According to an estimate, the mean income of attorneys was
$66,271 in 2015. A researcher wanted to check if the current mean
income of attorneys is greater than $66, 271. A random sample of 64
attorneys taken by this researcher produced a mean income of
$69,484 with a standard deviation of $11,500. Test at the 5 %
significant level whether the current mean income of all attorneys
is greater than $66,271. Explain your conclusion in words.

An engineer wanted to estimate the true mean resistance of a
certain electrical circuits (?) by a sample mean (?̅). It is known
that the population is normal, and the population standard
deviation is ? = 0.25 ohms. Determine the required sample size (?)
so that he will be 90% confident of being correct within ±
0.06.

Q: An agronomist wanted to estimate the mean
weight of a certain fruit (?) in his farm. He selected a random
sample of size ? = 16, and found that the sample mean ?̅ = 55 gm
and a sample stranded deviation ? = 3 gm. It is assumed that the
population is normal with unknown standard deviation (?).
a) Find a point estimate for (?).
b) Find a 95% confidence interval for the population mean
(?).

Suppose you are trying to estimate the population mean for the
number of Denver people that will vote in the next election. You
know there are 800,000 voters in Denver. You have sampled 41,000
Denver citizens and have calculated a sample mean of 150,000 likely
voters in the next election. You believe the population standard
deviation to be 80,000 voters. Please calculate a confidence
interval for your sample mean, assuming you wish to be 95%
confident.

Suppose the demand for crossing the Chargem Bridge is
given by Q = 10,000 – 1,000P where P is in $/car and Q is the
number of cars per day.If the toll (P) is $2/car, how much revenue is collected
daily?What is the price elasticity of demand at this point?Could the bridge authorities increase their revenues by
changing their price?The Crazy Canuck Lines, a ferry service that competes with the
Chargem Bridge begins operating hovercrafts that make commuting by
ferry...

An obstetrician wanted to determine whether the mean
number of births was the same for each of the five days of the
week. She randomly selected eight days for each of the five
weekdays and recorded the number of births on that day in the data
table below. Refer to the ANOVA results from Excel.
Monday
Tuesday
Wednesday
Thursday
Friday
10023
10265
10283
10456
10691
11189
11198
11465
11045
11621
11753
11944
12509
12577
12927
13521
11346
11084
11593
11570...

A travel magazine wanted to estimate the mean amount of leisure
time per week enjoyed by adults. The research department at the
magazine took a sample of 36 adults and obtained the following data
on the weekly leisure time (in hours):
15 12 18 23 11 21 16 13 9 19 26
14 7 18 11 15 23 26
10 8 17 21 12 7 19 21 11 13
21 16 14 9 15 12 10 14
a. What is...

1. A researcher wanted to estimate the mean
contributions made to charitable causes by all major companies. A
random sample of 18 companies produced by the following data on
contributions (in millions of dollars) made by them.
1.8, 0.6, 1.2, 0.3, 2.6, 1.9, 3.4, 2.6, 0.2
2.4, 1.4, 2.5, 3.1, 0.9, 1.2, 2.0, 0.8, 1.1
Assume that the contributions made to charitable
causes by all major companies have a normal distribution.
a. What is the point estimate for the population...

You construct two confidence intervals to estimate the mean
number of passengers on all air routes. For which sample below
would you expect a smaller margin of error (PLEASE CIRCLE the
appropriate letter and then briefly EXPLAIN)
1) A random sample of 50 air routes between Seattle, WA and
Portland, OR
2) A random sample of 50 air routes starting and ending in many
different cities across the USA

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