BUAL 2650 Final Exam Spring 2025 |
Questions and Answers (Rupee
Gupta)
Area Under Shaded Region - -P(a < x < b), (b-a)/(d-c)
- 3 types of distribution - -Uniform, Normal, Exponential
- Uniform distribution - -A continuous probability distribution curve that has
a rectangular shape, where the probability is evenly distributed over an
interval of numbers.
-F(x)= 1/d-c
- How to find the area under a curve using standard normal distribution - -
Z= (x - mu)/standard deviation
- Z-score - -A measure of how many standard deviations you are away from
the mean.
- Sample distribution - -A sampling distribution is a probability distribution
of data obtained from a larger number of samples that were taken from a
specific population. It is the distribution of frequencies from various different
outcomes that could possibly occur in a specific population.
- Basic properties of the sample mean - --The distribution of the population
of the sample mean looks roughly like a normal curve
-If normally distributed, for any sample size, n, the population of all possible
samples means is also normally distributed
-The mean of the population of all possible sample means is equal to the
mean of the population. (mu = mu(x-bar))
-The standard deviation of the sample population is less than one standard
deviation
- Central Limit Theorem - -If the sample size is sufficiently large, then the
sample means are approximately normally distributed. The sampling
distribution of the sample mean, X-bar> 30.
- The Empirical Rule - -68%, 95%, 99.7%
- Standard Deviation of a sample population - -standard deviation/ √n
, - If the population mean µ is exactly 31, what is the probability of observing
a sample mean that is greater than or equal to 31.56? n=50, standard
deviation= 0.8 - -Z= (31.56 - 31)/ (0.8/√50)= P(z>4.96)
The area under the curve is infinitely small (the Z chart does not go past 3.9)
- Suppose that a random variable x has a uniform distribution with c= 2 and
d=8. Find P(3 <=x <= 5). - -(5-3)/(8-2)= .333
- Find the area under the normal curve when P(Z >-2) - -1- .02275= .97725
- Find the area under the normal curve between P(1 < z < 2). -
-.97725- .84134= .1359
- In a manufacturing process, we are interested in measuring the average
length of a certain type of bolt. Past data indicate that the standard deviation
is .25 inches. How many manufactured bolts should be sampled in order to
make us 95% confident that the sample mean bolt length is within .02 inches
of the true mean bolt length? - -1) .02= Z(.95) x (.25/√n)
2) *find Z*
- 1-.95= .5/2= .025
- 1-.025- .97500= 1.96
3) .02= 1.96 x (.25/√n)
4) .02√n= 1.96 x .25
5) .02√n= .49
6)√n= .49/.02
7)√n= 24.5
8) n= 600.25
9) Sample population must be at least 601
- Standard Deviation - -A computed measure of how much scores vary
around the mean score
- Margin of error - -Z x (standard deviation/ √n)
- Correlation Coefficient - -1- alpha
Ex. 1- .05= .95
- The probability that the population mean is contained in the interval - -[x-
bar +- Z(a/2) x (standard deviation/ √n)]
- If standard deviation is unknown use - -- t= (x-bar- mu)/ (s/√n)
- n= [((ta/2) x S)/E]^2
- t distribution - -A distribution specified by degrees of freedom used to
model test statistics for the sample mean, differences between sample
means
Questions and Answers (Rupee
Gupta)
Area Under Shaded Region - -P(a < x < b), (b-a)/(d-c)
- 3 types of distribution - -Uniform, Normal, Exponential
- Uniform distribution - -A continuous probability distribution curve that has
a rectangular shape, where the probability is evenly distributed over an
interval of numbers.
-F(x)= 1/d-c
- How to find the area under a curve using standard normal distribution - -
Z= (x - mu)/standard deviation
- Z-score - -A measure of how many standard deviations you are away from
the mean.
- Sample distribution - -A sampling distribution is a probability distribution
of data obtained from a larger number of samples that were taken from a
specific population. It is the distribution of frequencies from various different
outcomes that could possibly occur in a specific population.
- Basic properties of the sample mean - --The distribution of the population
of the sample mean looks roughly like a normal curve
-If normally distributed, for any sample size, n, the population of all possible
samples means is also normally distributed
-The mean of the population of all possible sample means is equal to the
mean of the population. (mu = mu(x-bar))
-The standard deviation of the sample population is less than one standard
deviation
- Central Limit Theorem - -If the sample size is sufficiently large, then the
sample means are approximately normally distributed. The sampling
distribution of the sample mean, X-bar> 30.
- The Empirical Rule - -68%, 95%, 99.7%
- Standard Deviation of a sample population - -standard deviation/ √n
, - If the population mean µ is exactly 31, what is the probability of observing
a sample mean that is greater than or equal to 31.56? n=50, standard
deviation= 0.8 - -Z= (31.56 - 31)/ (0.8/√50)= P(z>4.96)
The area under the curve is infinitely small (the Z chart does not go past 3.9)
- Suppose that a random variable x has a uniform distribution with c= 2 and
d=8. Find P(3 <=x <= 5). - -(5-3)/(8-2)= .333
- Find the area under the normal curve when P(Z >-2) - -1- .02275= .97725
- Find the area under the normal curve between P(1 < z < 2). -
-.97725- .84134= .1359
- In a manufacturing process, we are interested in measuring the average
length of a certain type of bolt. Past data indicate that the standard deviation
is .25 inches. How many manufactured bolts should be sampled in order to
make us 95% confident that the sample mean bolt length is within .02 inches
of the true mean bolt length? - -1) .02= Z(.95) x (.25/√n)
2) *find Z*
- 1-.95= .5/2= .025
- 1-.025- .97500= 1.96
3) .02= 1.96 x (.25/√n)
4) .02√n= 1.96 x .25
5) .02√n= .49
6)√n= .49/.02
7)√n= 24.5
8) n= 600.25
9) Sample population must be at least 601
- Standard Deviation - -A computed measure of how much scores vary
around the mean score
- Margin of error - -Z x (standard deviation/ √n)
- Correlation Coefficient - -1- alpha
Ex. 1- .05= .95
- The probability that the population mean is contained in the interval - -[x-
bar +- Z(a/2) x (standard deviation/ √n)]
- If standard deviation is unknown use - -- t= (x-bar- mu)/ (s/√n)
- n= [((ta/2) x S)/E]^2
- t distribution - -A distribution specified by degrees of freedom used to
model test statistics for the sample mean, differences between sample
means
