For the GEOMETRIC distribution:
1a) Determine the most powerful critical region for testing H0 p=p0 against H1 p=θp (p1 > p0) using a random sample of size n.
1b) Find the uniformly most powerful H0 p<θ0 against H1 p>θ1
In: Statistics and Probability
For the Rayleigh distribution:
1a) Determine the most powerful critical region for testing H0 θ=θ0 against H1 θ=θ1 (θ1 > θ0) using a random sample of size n.
1b) Find the uniformly most powerful H0 θ<θ0 against H1 θ>θ1
In: Statistics and Probability
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(14.43) One reason why the Normal approximation may fail to give accurate estimates of binomial probabilities is that the binomial distributions are discrete and the Normal distributions are continuous. That is, counts take only whole number values, but Normal variables can take any value. We can improve the Normal approximation by treating each whole number count as if it occupied the interval from 0.5 below the number to 0.5 above the number. For example, approximate a binomial probabilityP(X⩾10)P(X⩾10)by finding the Normal probabilityP(X⩾9.5)P(X⩾9.5). Be careful: binomialP(X>10)P(X>10)is approximated by NormalP(X⩾10.5)P(X⩾10.5). Adding 0.5 to the length of the interval is called continuity correction. One statistic used to assess professional golfers is driving accuracy, the percent of drives that land in the fairway. In 2013, driving accuracy for PGA Tour professionals ranged from about 45% to about 75%. Tiger Woods, the highest money winner on the PGA tour in 2013, only hits the fairway about 57 % of the time. We will assume that his drives are independent and that each has probability 0.57 of hitting the fairway. Suppose Woods drives 24 times. (a)Does this setting satisfies the rule of thumb for use of the Normal approximation (just barely)
(b) The exact binomial probability that he hits 17 or more fairways is (±±0.001) (c) What is the Normal approximation (±±0.0001 (use software)) to P(X⩾17)P(X⩾17) without using continuity correction? (d) What is the Normal approximation (±±0.0001 (use software)) to P(X⩾17)P(X⩾17) using the continuity correction? |
In: Statistics and Probability
Given a normal distribution with µ = 47 and σ = 6, what is the probability that:
X < 39 or X > 44
X is between 37 and 46
7% of the values are less than what X value.
Between what two X values (symmetrically distributed around the mean) are 70% of the values?
In: Statistics and Probability
Health care issues are receiving much attention in both academic and political arenas. A sociologist recently conducted a survey of citizens over 60 years of age whose net worth is too high to qualify for Medicaid. The ages of 25 senior citizens were as follows: 60 65 71 76 86 61 66 73 77 87 62 68 73 81 89 63 68 74 81 90 64 69 75 82 95
Calculate the arithmetic mean age of the senior citizens. (Note: Use two decimal places, i.e. 12.34)
Determine the median of the senior ages.
Determine the first quartile of the ages of the senior citizens.
Determine the third quartile of the ages of the senior citizens.
Determine the interquartile range of the ages of the senior citizens.
Compute the skewness.
Compute the Kurtosis.
Calculate the variance. (Note: Use two decimal places, i.e. 12.34)
Calculate standard deviation. (Note: Use two decimal places, i.e. 12.34%)
Calculate the coefficient of variation.
In: Statistics and Probability
An investigator predicts that dog owners in the country spend more time walking their dogs than do dog owners in the city. The investigator gets a sample of 21 country owners. The mean number of hours per week that city owners (known population) spend walking their dogs is 10.0. The mean number of hours country owners spent walking their dogs per week was 15.0. The standard deviation of the number of hours spent walking the dog by owners in the country was 4.0. Do dog owners in the country spend more time walking their dogs than do dog owners in the city? Please test the investigator's theory using an alpha level of .05.
For each of the single sample t-tests below (problem 1-3),
please include:
1) The null and alternative hypotheses (can be written in notation
or as a sentence)
2) Calculate (show all work):
a) The estimated population variance
b) The variance of the distribution of means
c) The standard deviation of the distribution of means
3) Degrees of freedom
4) The cutoff sample score on the comparison distribution at which
the null hypothesis
should be rejected (cutoff scores)
5) The sample’s score on the comparison distribution (e.g.
t-obtained) (show all work)
6) Decide whether to reject the null hypothesis
7) State your conclusion in APA format (make sure to report what
groups you are comparing
(and their means), what the DV is, whether it was significant or
not and your test statistic (t)
(see the lecture slides for examples).
In: Statistics and Probability
The lengths of a particular animal's pregnancies are approximately normally distributed, with mean
mu
equals280days and standard deviation
sigma
equals12
days.
(a) What proportion of pregnancies lasts more than
301
days?
(b) What proportion of pregnancies lasts between
262
and
283
days?
(c) What is the probability that a randomly selected pregnancy lasts no more than
256
days?
(d) A "very preterm" baby is one whose gestation period is less than
250
days. Are very preterm babies unusual?
LOADING...
Click the icon to view a table of areas under the normal curve.
(a) The proportion of pregnancies that last more than
301
days is
nothing
.
(Round to four decimal places as needed.)
(b) The proportion of pregnancies that last between
262
and
283
days is
nothing
.
(Round to four decimal places as needed.)
(c) The probability that a randomly selected pregnancy lasts no more than
256
days is
nothing
.
(Round to four decimal places as needed.)
(d) A "very preterm" baby is one whose gestation period is less than
250
days. Are very preterm babies unusual?
The probability of this event is
nothing
,
so it
▼
would not
would
be unusual because the probability is
▼
greater
less
than 0.05.
(Round to four decimal places as needed.)
In: Statistics and Probability
A certain flight arrives on time
90
percent of the time. Suppose
171
flights are randomly selected. Use the normal approximation to the binomial to approximate the probability that
(a) exactly
152
flights are on time.
(b) at least
152
flights are on time.
(c) fewer than
143
flights are on time.
(d) between
143
and
153
,
inclusive are on time.
(a)
P(152
)equalsnothing
(Round to four decimal places as needed.)
(b)
P(Xgreater than or equals
152)equalsnothing
(Round to four decimal places as needed.)
(c)
P(Xless than
143)equalsnothing
(Round to four decimal places as needed.)
(d)
P(143
less than or equalsXless than or equals153)equalsnothing(Round to four decimal places as needed.)
In: Statistics and Probability
14. For a sample of students, their total SAT scores and college grade point averages (GPA) are given below.
SAT GPA
1070 3.50
1040 2.66
1120 3.54
1260 3.83
1100 2.60
1020 2.70
960 2.00
a. Make a scatterplot on your calculator. Does there appear to be a linear relationship between SAT scores and GPA?
b. Find the least squares regression line. Round all constants to 4 decimal places.
c. Use your equation to predict the college GPA of a student with a 1000 SAT score.
d. Find the correlation coefficient, rounded to 3 decimal places. What does this tell you about the relationship between the two variables?
In: Statistics and Probability
Let X1,..., Xn be a random sample from the Bernoulli distribution:
Use the likelihood ratio test to give a form of test (without specifying the value of the critical value) for H0: p= 1 versus H1:p ≠1.
In: Statistics and Probability
Inorganic phosphorous is a naturally occurring element in all plants and animals, with concentrations increasing progressively up the food chain (fruit < vegetables < cereals < nuts < corpse). Geochemical surveys take soil samples to determine phosphorous content (in ppm, parts per million). A high phosphorous content may or may not indicate an ancient burial site, food storage site, or even a garbage dump. The Hill of Tara is a very important archaeological site in Ireland. It is by legend the seat of Ireland's ancient high kings†. Independent random samples from two regions in Tara gave the following phosphorous measurements (ppm). Assume the population distributions of phosphorous are mound-shaped and symmetric for these two regions.
| Region I: x1; n1 = 12 | |||||
| 540 | 810 | 790 | 790 | 340 | 800 |
| 890 | 860 | 820 | 640 | 970 | 720 |
| Region II: x2; n2 = 16 | |||||||
| 750 | 870 | 700 | 810 | 965 | 350 | 895 | 850 |
| 635 | 955 | 710 | 890 | 520 | 650 | 280 | 993 |
(a) Use a calculator with mean and standard deviation keys to calculate x1, s1, x2, and s2. (Round your answers to one decimal place.)
| x1 | = |
| s1 | = |
| x2 | = |
| s2 | = |
(b) Let μ1 be the population mean for
x1 and let μ2 be the
population mean for x2. Find a 90% confidence
interval for μ1 − μ2.
(Round your answers to one decimal place.)
| lower limit | |
| upper limit |
| Region I: x1; n1 = 15 | |||||||
| 855 | 1550 | 1230 | 875 | 1080 | 2330 | 1850 | 1860 |
| 2340 | 1080 | 910 | 1130 | 1450 | 1260 | 1010 | |
| Region II: x2; n2 = 14 | |||||||
| 540 | 810 | 790 | 1230 | 1770 | 960 | 1650 | 860 |
| 890 | 640 | 1180 | 1160 | 1050 | 1020 | ||
(c) Use a calculator with mean and standard deviation keys to verify that x1, s1, x2, and s2. (Round your answers to one decimal place.)
| x1 | = |
| s1 | = |
| x2 | = |
| s2 | = |
(d) Let μ1 be the population mean for
x1 and let μ2 be the
population mean for x2. Find a 90% confidence
interval for μ1 − μ2.
(Round your answers to one decimal place.)
| lower limit | |
| upper limit |
In: Statistics and Probability
12. A major study of alternative welfare programs randomly assigned women on welfare to one of two programs, called “WIN” and “Options”. WIN was the existing program. The new Options program gave more incentives to work. An important question was how much more (on the average) women in Options earned than those in WIN. Here is data for earnings in dollars over a three-year period:
Program Number of Participants Average Earnings Standard Deviation
Options 1362 7638 289
WIN 1395 6595 247
Is there evidence at the 5% significance level that the Options participants earn significantly more (on average) than those in the WIN program?
In: Statistics and Probability
10. The National Collegiate Athletic Association (NCAA) requires colleges to report the graduation rates of their athletes. Here are data from a Big Ten university's report. For parts a and b, state your hypotheses, the test statistic and p-value, and then write a conclusion in terms of the problem.
(a) Ninety-five of the 147 athletes admitted in 1989-1991 graduated within six years. Did the percent of athletes who graduated within six years differ significantly from the all-university percentage, which was 70%?
(b) The graduation rates were 37 of 45 female athletes and 58 of 102 male athletes. Is there evidence that a smaller proportion of male athletes than of female athletes graduated within six years?
In: Statistics and Probability
A resort that is a major vacation destination consists of 1,450 acres of land, so the resort provides shuttle buses for visitors who need to travel within the resort. Suppose the wait time for a shuttle bus follows a uniform distribution with a minimum time of 30 seconds and a maximum time of 9 minutes. Complete parts a through e.
a. What is the probability that a visitor will need to wait more than 22 minutes for the next shuttle?
The probability is _________.
(Round to four decimal places as needed.)
b. What is the probability that a visitor will need to wait less than 6.5 minutes for the next shuttle?
The probability is _______.
(Round to four decimal places as needed.)
c. What is the probability that a visitor will need to wait between 5 and 7 minutes for the next shuttle?
The probability is ________.
d. Calculate the mean and standard deviation for this distribution.
The mean of the given uniform distribution is μ=______. (Type an integer or a decimal. Do not round.)
The standard deviation of the given uniform distribution is σ =________. (Round to four decimal places as needed.)
d. Calculate the mean and standard deviation for this distribution.
The mean of the given uniform distribution is μ=_______.
(Type an integer or a decimal. Do not round.)
The standard deviation of the given uniform distribution is
σ=_______. (Round to four decimal places as needed.)
e. The resort has a goal that 70% of the time, the wait for the shuttle will be less than 66 minutes. Is this goal being achieved?
The goal (is;is not) being achieved, because the area under the distribution to the left of 66 is _______
(Round to four decimal places as needed.)
In: Statistics and Probability
For the Rayleigh distribution:
1a) Determine the most powerful critical region for testing H0 θ=θ0 against H1 θ=θ1 (θ1 > θ0) using a random sample of size n.
1b) Find the uniformly most powerful H0 θ<θ0 against H1 θ>θ1
In: Statistics and Probability