The mean and the std. dev. of the sample mean. Select a SRS of size n from a population and measure a variable X on each individual in the sample. The data consists of observations on n r.v’s X 1 ,X 2 …,X n . If the population is large we can consider X 1 ,X 2 …,X n to be independent.
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Answer: μT = μ X1+X2+···+Xn = n·μ
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A bottling company uses a filling machine to fill plastic bottles with a popular cola. The bottles are supposed to contain 300 milliliters (ml). In fact, the contents vary according to a normal distribution with mean 298 ml and standard deviation 3 ml.
(a) What is the probability that an individual bottle contains less
than 295 ml?
(b) What is the probability that the mean contents of the bottles in
a six-pack is less than 295ml?
.
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is approximately normal with mean and std dev. / .
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Suppose that the weights of airline passengers are known to have a distribution with a mean of 75kg and a std. dev. of 10kg. A certain plane has a passenger weight capacity of 7700kg. What is the probability that a flight of 100 passengers will exceed the capacity?
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In a certain University, the course STA100 has tutorials of size 40. The course STA200 has tutorials of size 25, and the course STA300 has tutorials of size 15. Each course has 5 tutorials per year. Students are enrolled by computer one by one into tutorials. Assume that each student being enrolled by computer may be considered a random selection from a very big group of people wherein there is a 50-50 male to female sex ratio. Which of the following statements is true?
A) Over the years STA100 will have more tutorials with 2/3 females (or more).
B) Over the years STA200 will have more tutorials with 2/3 females (or more).
C) Over the years STA300 will have more tutorials with 2/3 females (or more).
D) Over the years, each course will have about the same number of tutorials with 2/3 females (or more).
E) No course will have tutorials with 2/3 females (or more).
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State whether the following statements are true or false.
(i) As the sample size increases, the mean of the sampling distribution of the sample mean decreases.
(ii) As the sample size increases, the standard deviation of the sampling distribution of the sample mean decreases.
(iii) The mean of a random sample of size 4 from a negatively skewed distribution is approximately normally distributed.
(iv) The distribution of the proportion of successes in a sufficiently large sample is approximately normal with mean p and standard deviation where p is the population proportion and n is the sample size.
(v) If is the mean of a simple random sample of size 9 from N(500, 18) distribution, then has a normal distribution with mean 500 and variance 36.
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State whether the following statements are true or false.
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A parking lot is patrolled twice a day (morning and afternoon). In the morning, the chance that any particular spot has an illegally parked car is 0.02. If the spot contained a car that was ticketed in the morning, the probability the spot is also ticketed in the afternoon is 0.1. If the spot was not ticketed in the morning, there is a 0.005 chance the spot is ticketed in the afternoon.
a) Suppose tickets cost $10. What is the expected value of the tickets for a single spot in the parking lot.
b) Suppose the lot contains 400 spots. What is the distribution of the value of the tickets for a day?
c) What is the probability that more than $200 worth of tickets are written in a day?
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P(c < X < d ) = 0.95.
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P( c < < d ) = 0.95
8. X~ N(, ) Let be the mean of a random sample of size n
Find the values of c and d such that P( c < < d ) = 0.95
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Margin of error = m =
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is the stdev. of the sample mean (this is also known as the std. error of the sample mean ) and it can also be written as
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A limnologist wishes to estimate the mean phosphate content per unit volume of lake water. It is known from previous studies that the stdev. has a fairly stable value of 4mg. How many water samples must the limnologist analyze to be 90% certain that the error of estimation does not exceed 0.8 mg?
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DRP Scores
40 26 39 14 42 18 25 43 46 27 19
47 19 26 35 34 15 44 40 38 31 46
52 25 35 35 33 29 34 41 49 28 52
47 35 48 22 33 41 51 27 14 54 45
Z Confidence Intervals
The assumed sigma = 11.0
Variable N Mean StDev SE Mean 95.0 % CI
DRP Scor 44 35.09 11.19 1.66 (31.84 , 38.34)
Stat > Basic Statistics >1 Sample Z and select ‘Confidence interval’
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A random sample of 85 students in Chicago city high schools taking a course designed to improve SAT scores. Based on these students a 90% CI for the mean improvement in SAT scores for all Chicago high school students is computed as (72.3, 91.4) points.
Which of the following statements are true?
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The statement being tested in a test of significance is called the null hypothesis. The test of significance is designed to assess the strength of the evidence against the null hypothesis. Usually the null hypothesis is a statement of “no effect” or “no difference”.
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Each of the following situations requires a significance test about a population mean . State the appropriate null hypothesis H0 and alternative hypothesis Ha in each case.
(b) Larry's car consume on average 32 miles per gallon on the highway. He now switches to a new motor oil that is advertised as increasing gas mileage. After driving 3000 highway miles with the new oil, he wants to determine if his gas mileage actually has increased.
(c) The diameter of a spindle in a small motor is supposed to be 5 millimeters. If the spindle is either too small or too large, the motor will not perform properly. The manufacturer measures the diameter in a sample of motors to determine whether the mean diameter has moved away from the target.
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H0: μ = 10 ;
Ha: μ > 10 .
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If the P-value is as small or smaller than , we reject H0 and say that the data are statistically significant at level .
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Ha : µ > µ0 is P( Z ≥ z )
Ha : µ < µ0 is P( Z ≤ z )
Ha : µ ≠ µ0 is 2·P( Z ≥ |z|)
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Z-Test
Test of mu = 32.00 vs mu > 32.00
The assumed sigma = 11.0
Variable N Mean StDev SE Mean Z P
DRP Scor 44 35.09 11.19 1.66 1.86 0.031
Stat > Basic Statistics >1 Sample Z and select ‘Test mean’
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For the exercise on slide 27 a 95% CI is
83 ± 1.96·(7/5) = (80.256, 85.744)
The value 80 is not in this interval and so we reject H0: = 80 at the 5% level of significance.
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