In: Statistics and Probability

**For a normal population with known variance
σ ^{2}, what**

** **
*x**-**z**α**/2**σ**n**≤**μ**≤**x**+**z**α**/2**σ**n*

solution:

At 98% confidence level the z is ,

= 1 - 98% = 1 - 0.98 = 0.02

/ 2 = 0.02/ 2 = 0.01

Z/2 = Z0.01 = 2.326 ( Using z table )

Margin of error = E = Z/2
* (
/n)

At 98% confidence interval

is,

- E < < + E

- Z/2 * ( /n) < < + Z/2 * ( /n)

- 2.326 * ( /n) < < + 2.326 * ( /n)

For a normal population with known variance σ2, answer the
following questions:
d. What is the conﬁdence level for the interval μ ≤ x+ 2.00σ∕ √
n?
e. What is the conﬁdence level for the interval x−1.96σ∕ √ n ≤
μ?

Assume a normal population with known variance σ2, a random
sample (n< 30) is selected. Let x¯,s represent the sample mean
and sample deviation. (1)(2pts) write down the formula: 98%
one-sided confidence interval with upper bound for the
population mean. (2)(6pts) show how to derive the confidence
interval formula in (1).

For a normal population with known variance (sigma)^2, what is
the confidence level for the CI
(mean)-2.14(sigma)/√n≤μ≤(mean)+2.14(sigma)/√n ?

For a normal population with known variance s2, answer the
following questions:
(a) What is the confidence level for
the interval: ?− 1.85?/√?≤ ?≤ ?+ 1.85?/√??
(b) What is the confidence level for
the interval: ?≤ ?+ 1.4?/√?

Consider a normal population distribution with the value of \(\sigma\) known.(a) What is the confidence level for the interval \(\bar{x} \pm 2.88 \sigma / \sqrt{n} ?\) (Round your answer to one decimal place.)\(\%\)(b) What is the confidence level for the interval \(\bar{x} \pm 1.47 \sigma / \sqrt{n} ?\) (Round your answer to one decimal place.) \(\%\)(c) What value of \(z_{\alpha / 2}\) in the CI formula below results in a confidence level of \(99.7 \% ?\) (Round your answer to...

Suppose we take a random sample X1,…,X5 from a normal population
with unknown variance σ2 and unknown mean μ. We test the
hypotheses
H0:σ=2 vs. H1:σ<2
and we use a critical region of the form {S2<k} for some
constant k.
(1) - Determine k so that the type I error probability α of the
test is equal to 0.05.
(2) - For the value of k found in part (1), what is your
conclusion in this test if you see...

Suppose we take a random sample X1,…,X5 from a normal population
with an unknown variance σ2 and unknown mean μ.
Construct a two-sided 95% confidence interval for σ2 if the
observations are given by
−1.25,1.91,−0.09,2.71,2.70

Consider a normal population distribution with the value of σ known.(a) What is the confidence level for the interval \(\bar{x} \pm 2.81 \sigma / \sqrt{n} ?\) (Round your answer to one decimal place.) \(\%\)(b) What is the confidence level for the interval \(\bar{x} \pm 1.43 \sigma / \sqrt{n} ?\) (Round your answer to one decimal place.) \(\%\)(c) What value of \(z_{\alpha / 2}\) in the CI formula below results in a confidence level of \(99.7 \% ?\) (Round your answer...

Use technology to construct the confidence intervals for the
population variance σ2 and the population standard deviation σ.
Assume the sample is taken from a normally distributed
population.
c=0.90
s=33
n=20
The confidence interval for the population variance is (_, _ )
(Round to two decimal places as needed.)
The confidence interval for the population standard deviation is
(_, _ ) (Round to two decimal places as needed.)

Use technology to construct the confidence intervals for the
population variance σ2 and the population standard deviation σ2
Assume the sample is taken from a normally distributed
population.
c=0.980.98,
s2=4.844.84,
n=2727
The confidence interval for the population variance is
(Round to two decimal places as needed.)

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