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Test the following hypotheses (assuming that XDX_{D} is normally distributed): ...
Oct 11, 2024
Solution by Steps
step 1
Calculate the test statistic for set "a" using the formula: t=XˉD−μDsDnD t = \frac{\bar{X}_D - \mu_D}{\frac{s_D}{\sqrt{n_D}}} where XˉD=2 \bar{X}_D = 2 , μD=0 \mu_D = 0 , sD=4 s_D = 4 , and nD=15 n_D = 15
step 2
Substitute the values into the formula for set "a": t=2−0415 t = \frac{2 - 0}{\frac{4}{\sqrt{15}}}
step 3
Calculate the test statistic for set "b" using the same formula: t=XˉD−μDsDnD t = \frac{\bar{X}_D - \mu_D}{\frac{s_D}{\sqrt{n_D}}} where XˉD=−8 \bar{X}_D = -8 , μD=0 \mu_D = 0 , sD=20 s_D = 20 , and nD=50 n_D = 50
step 4
Substitute the values into the formula for set "b": t=−8−02050 t = \frac{-8 - 0}{\frac{20}{\sqrt{50}}}
step 5
Determine the critical t-value for set "a" from the t-distribution table using α=0.05 \alpha = 0.05 and df=nD−1=15−1 df = n_D - 1 = 15 - 1
step 6
Determine the critical t-value for set "b" from the t-distribution table using α=0.01 \alpha = 0.01 and df=nD−1=50−1 df = n_D - 1 = 50 - 1
step 7
Compare the calculated t-statistics with the critical t-values to decide whether to reject or fail to reject the null hypotheses for both sets "a" and "b"
a Answer
[Insert final answer for set "a" here]
b Answer
[Insert final answer for set "b" here]
Key Concept
Hypothesis testing using t-statistics
Explanation
The t-statistic is used to determine whether there is a significant difference between the sample mean and the hypothesized population mean when the sample size is small and the population standard deviation is unknown.
A statement made about a population for testing purpose is called? A StatisticB HypothesisC Level of SignificanceD Test-Statistic
Solution by Steps
step 1
Identify the correct term that describes a statement made about a population for testing purposes
Answer
Hypothesis
Key Concept
Hypothesis
Explanation
A hypothesis is a statement made about a population parameter that can be tested using statistical methods. It is an assumption or claim about the characteristic of a population, such as the population mean (μ\mu), which can be tested against observed data.
Multiple Choice: A statement made about a population for testing purpose is called? A Statistic B Hypothesis C Level of Significance D Test-Statistic
Solution by Steps
step 1
Identify the correct term that describes a statement made about a population for testing purposes
Answer
B
Key Concept
Hypothesis
Explanation
A hypothesis is a statement made about a population parameter that can be tested using statistical methods.
Multiple Choice: To test a hypothesis involving proportions, both np and n(1-p) should A Be at least 30 B Be greater than 5 C Lie in the range from 0 to 1 D Be greater than 50
Solution by Steps
step 1
To ensure the normal approximation to the binomial distribution is appropriate, both np np and n(1−p) n(1-p) must satisfy a certain condition
step 2
The rule of thumb for the normal approximation to be valid is that both np np and n(1−p) n(1-p) should be greater than 5
∻Answer∻
The correct answer is that both np np and n(1−p) n(1-p) should be greater than 5
B
Key Concept
Normal Approximation Validity Conditions
Explanation
For the normal approximation to the binomial distribution to be valid, both np np and n(1−p) n(1-p) must be greater than 5.
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