Answered: Let data be x1, . . . , xn. each…

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter8: Sequences, Series, And Probability

Section8.CT: Chapter Test

Problem 16CT: How many six-digit license plates can be made if no plate begins with 0 or 1?

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Question

Let data be x1, . . . , xn. each data-point xi is normally distributed, with unknown mean θ and standard deviation assumed to be equal to 1.
As a prior distribution for θ, we assign a normal distribution with mean 5 and standard deviation 10.
(a) What is the posterior distribution for θ? (b) What is the posterior mean for θ?
(c) What is the posterior median for θ?
(d) What is a 95% equal tail credible interval for θ?

X
-.136
.246
1.946
-.786
1.783
1.234
-.02
-.391
-.068
-.103
.718
1.185
-.577
-.292
1.414
1.55
.93
1.397
1.469
1.842
-.351
2.271
-.666
1.006
.499
1.396
.933
-.177
1.656
.762
.709
1.161
.702
.845
-.31
3.184
.47
.437
1.003
1.45
1.836
.867
.163
.867
.541
-.564
-.474
1.464
-1.185
-.37

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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