Answered: Eastland and Westland are two…

A First Course in Probability (10th Edition)

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ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

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Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Question

Eastland and Westland are two neighboring countries. A random sample of 101 families from Eastland has a sample mean income of $52 thousand and a sample standard deviation $10.3 thousand.

a. Find and interpret a 95% confidence interval for the population mean income in Eastland?

An independent sample of 31 families from Westland has sample mean income of $48 thousand and a sample standard deviation of 11.6 thousand.

b. Test the null hypothesis that the population mean income in Eastland equals the population mean income in Westland against a two-sided alternative. Include in your answer the null and alternative hypotheses. Show calculations and state the conclusion.

c. Find a 90% confidence interval for the difference between the population mean income in Eastland and the population mean income in Westland

d. Clearly describe what happens to the interval estimate in (c) when the sample size increases.

e. It is claimed that mean income in Eastland equals the population mean income in Westland. Using your results in (c). do you reject this claim? Clearly explain why. Is your answer consistent with your conclusion in (b)? Explain.

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 + ...

Expert Solution

Step 1

Comments : As per the company norms we have to solved first three parts only.

Solution:

Given information:

$n_{1} = 101 S a m p l e s i z e o f E a s t l a n d n_{2} = 31 S a m p l e s i z e o f W e s t l a n d x_{1} = $ 52 S a m p l e m e a n o f E a s t l a n d x_{2} = $ 48 S a m p l e m e a n o f W e s t l a n d s_{1} = $ 10.3 S a m p l e s \tan d a r d d e v i a t i o n s o f E a s t l a n d s_{2} = $ 11.6 S a m p l e s \tan d a r d d e v i a t i o n s o f W e s t l a n d α = 0.05 L e v e l o f s i g n i f i c a n c e$

To find pooled variance

${s_{p}}^{2} = \frac{(n_{1} - 1) \times {s_{1}}^{2} + (n_{2} - 1) \times {s_{2}}^{2}}{n_{1} + n_{2} - 2} {s_{p}}^{2} = \frac{(101 - 1) \times 10 . 3^{2} + (31 - 1) \times 11 . 6^{2}}{101 + 31 - 2} {s_{p}}^{2} = \frac{100 \times 106.09 + 30 \times 134.56}{132 - 2} {s_{p}}^{2} = \frac{10609 + 4036.8}{130} {s_{p}}^{2} = \frac{14645.8}{130} {s_{p}}^{2} = 112.66$

Step 2

a) The 95% confidence interval for the population mean income in Eastland is

$x_{1} \pm t_{α / 2, n - 1} \times \frac{s_{1}}{\sqrt{n_{1}}} A t α = 0.05 t_{α / 2, n - 1} = t_{0.025, 100} = 1.984 F r o m e x c e l (52 \pm 1.984 \times \frac{10.3}{\sqrt{101}}) (52 \pm 1.984 \times \frac{10.3}{10.049875}) (52 \pm 1.984 \times 1.0248883) (52 \pm 2.0333785) (49.96662, 54.033378) (49.9666, 54.0334)$

The 95% confidence interval for the population mean income in Eastland is 49.9666 to 54.0334

Interpretation of confidence interval:

We can be 95% confident that the true population mean income in Eastland is between 49.9666 to 54.0334

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