The notation X ~ N (µ, σ) means that the RV X has a normal
distribution with mean µ and standard deviation σ. USE A RULER FOR
ALL DRAWINGS.
1. What is the z-score of x, when x = 1, and X ~ N (12, 3)?
2. The average score on a math test was 70 points with a
standard deviation of 12 points. Jane’s z-score was 2.2. How many
points did she score?
3. In 2009, Guinness World Records...
(a) Suppose X1, ..., Xn1 i.i.d. ∼ N(µ1, σ2 1 ) and Y1, ...Yn2
i.i.d. ∼ N(µ2, σ2 2 ) are independent Normal samples. Suggest an
unbiased estimator for µ1 − µ2 and find its’ standard error. Now
Suppose n1 = 100, n2 = 200, it is known that σ 2 1 = σ 2 2 = 1, and
we calculate X¯ = 5.7, Y¯ = 5.2. Find a 2-standard-error bound on
the error of estimation. (b) Suppose X ∼...
Suppose X1, ..., Xn are i.i.d. from an exponential distribution
with mean θ. If we are testing H0 : θ = θ0 vs
Ha : θ > θ0. Suppose we reject H0 when
( X¯n/ θ0) > 1 + (1.645/
√n)
(a) (10 points) Calculate the power function G(ζ). You may leave
your answer in terms of the standard normal cdf Φ(x).
(b) (5 points) Is this test consistent?
A sample of n = 36 observations is drawn from a normal
population, with = 500 and = 42. Find P(500
< X̅ < 510).
a.
0.0764
b.
0.8756
c.
0.9236
d.
0.7556
e.
0.4236
7.A random sample of 49 observations is drawn from a population
with a normal distribution. If the sample mean is 265 and the
sample standard deviation is 57, find the 95% confidence interval
for the population mean.
8. A random sample of 49 observations is drawn from a population
with a normal distribution with a standard deviation of 57. If the
sample mean is 265, find the 95% confidence interval for the
population mean. Compare this confidence interval with the...
Suppose X1; : : : ; Xn is i.i.d Exponential distribution with
density
f(xjθ) = (1/θ) * e(-x/θ); 0 ≤ x < 1; θ > 0:
(a) Find the UMVUE (the best unbiased estimator) of θ.
(b) What is the Cramer-Rao lower bound of all unbiased estimator of
all unbiased estimator
of θ. Does the estimator from (a) attain the lower bound? Justify
your answer.
(c) What is the Cramer-Rao lower bound of all unbiased estimator of
θ^2?
3
(d)...
Suppose you have selected a random sample of n =5 measurements
from a normal distribution. Compare the standard normal z values
with the corresponding t values if you were forming the following
confidence intervals.
(a) 8080% confidence interval
z=
t=
(b) 9898% confidence interval
z=
t=
(c) 9595% confidence interval
z=
t=
ConfidenceIntervals
Section 8.1 #38
A sample of size n = 80 is drawn from a
normal population whose standard deviation is σ = 6.8.
The sample mean is x̄ = 40.41.
a.
Construct a 90% confidence interval for
μ.
b.
If the population were not approximately normal,
would the confidence interval constructed in part (a) be valid?
Explain
A sample of size n =52 is drawn from a normal population whose
standard deviation is σ=7.9. The sample mean is x=43.78
(a) Construct a 80% confidence interval for μ. Round the answer
to at least two decimal places..
(b) If the population were not approximately normal, would the
confidence interval constructed in part (a) be valid? Is the sample
large?