**Title: Conducting Hypothesis Tests Using the T-Distribution****Introduction:**When conducting hypothesis tests using small sample sizes, it is essential to verify the normality assumption of the sample distributions. This ensures the accuracy of the t-distribution method for hypothesis testing. In this example, we will test the given hypotheses using provided sample results.**Hypotheses:**- Null Hypothesis (\(H_0\)): \(\mu_1 = \mu_2\)- Alternative Hypothesis (\(H_a\)): \(\mu_1 > \mu_2\)**Sample Data:**- Sample 1: - Mean (\(\bar{x}_1\)): 56 - Standard deviation (\(s_1\)): 8.2 - Sample size (\(n_1\)): 30- Sample 2: - Mean (\(\bar{x}_2\)): 51 - Standard deviation (\(s_2\)): 6.9 - Sample size (\(n_2\)): 40**Task:**(a) Calculate the test statistic and the p-value.- Enter the test statistic rounded to two decimal places.- Enter the p-value rounded to three decimal places.**Interactive Elements:**- Textbox to input the test statistic.- Textbox to input the p-value.- Button for additional resources: eTextbook and Media.- Button for hints: Hint.(b) **Conclusion of the Test:**Determine the result of the test at a significance level of 5%.- Options for conclusion: - Reject \(H_0\). - Do not reject \(H_0\).Use this example to understand the application of the t-distribution in hypothesis testing and how to make decisions based on statistical analysis.

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Answered: Use the t-distribution and the given…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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We want to conduct a hypothesis test of the claim that the population mean time it takes drivers to react following the application of brakes by the driver in front of them is less than 2.2 seconds. So, we choose a random sample of reaction time measurements. The sample has a mean of 2.3 seconds and a standard deviation of 0.5 seconds. For each of the following sampling scenarios, choose an appropriate test statistic for our hypothesis test on the population mean. Then calculate that statistic. Round your answers to two decimal places. (a) The sample has size 12, and it is from a population with a distribution about which we know very little. O z = O t = O It is unclear which test statistic to use. (b) The sample has size 11, and it is from a normally distributed population with a known standard deviation of 0.40. O z = O t = O It is unclear which test statistic to use.
Suppose you needed to test the claim that the two samples described below come from populations with the same mean. Assume that the samples are independent simple random samples. Sample 1: n1 = 13, a1 = 24, s1 = 7.55 Sample 2: n2 = 3, x2 = 21.9, s2 = 8.92 Find: (a) The estimated degree of freedom is (b) The standardized test statistic is (use Sample 1 - Sample 2)
Suppose we have two normal populations denoted A and B, and we want to compare their population variances by estimating the ratio: variance of population B divided by variance of population A. We have an independent random sample of size 15 from population A and an independent random sample of size 9 from population B. The sample variance for population B turns out to be 30 percent larger than that for population A. What is the 99 percent confidence interval for the ratio: population variance of population B divided by population variance of population A? Use stattrek.org for finding the needed critical values.
7. What is the 95% confidence interval for the difference in the population means, µ1 - µ2, given the data from the two independent samples: n1 =56, x1= 58.2, s1= 4.3, n2=45, x2=60.2, s2=5.5 B. [-3.28 , -0.72] C. [-3.60, -0.40] D. [-3.65 , -0.35] A. [-3.96 , -0.04] 8. What is the 98% confidence interval for the difference in the population means, µ1 - µ2, given: n1= 22, x1= 41.4, s1= 5.9, n2= 19, x2= 43.2, s2= 6.5 D. [-3.48 , -0.12] B. [-3.07 , -0.53] A. [-3.12, -0.48] C. [-3.18, -0.42]
Katie thinks that people living in a rural environment have a healthier lifestyle than other people. She believes the average lifespan in the USA is 77 years and the lifespans follow an approximately normal distribution. A random sample of 13 obituaries from newspapers from rural towns in Idaho give = 78.9 and s = 2.83. Does this sample provide evidence that people living in rural Idaho communities live longer than 77 years? (a) State the null and alternative hypotheses: (Type "mu" for the symbol μ, e.g. mu >1 for the mean is greater than 1, mu < 1 for the mean is less than 1, mu not = 1 for the mean is not equal to 1) Ho: Ha (b) Find the test statistic = (c) Answer the question: Does this sample provide evidence that people living in rural Idaho communities live longer than 77 years? (Use a 10% level of significance) (Type: Yes or No)
(2) Choose the one alternative that best completes the statement or answers the question. Assume two independent random samples are available which provide sample proportions. For the first sample assume n₁ = 100 and x₁= 39. For the second sample, assume n₂= 100 and x₂= 49. Test the null hypothesis that the population proportions are equal versus the alternative hypothesis that the proportions are not equal at the 90% confidence level. Frame the test statistic by subtracting the proportion for population 1 from that for population 2. Pick an appropriate z value, p-value and conclusion. Round your answer to the nearest thousandth. Note that this is a two- tailed test. z-value = -1.425 p-value= 0.077 statistically significant z-value = 1.425 p-value= 0.077 statistically significant z-value = -1.425 p-value= 0.077 not statistically significant z-value = 1.425 p-value= 0.154 statistically not significant
Let Y₁, Y₂, ..., Y₁ denote a random sample from a normal distribution with unknown mean and unknown variance o². For testing Ho: o² = 0 vs. Ha: o²o, show that the a-level Likelihood Ratio Test (LRT) is equivalent to the x² test
Use the t-distribution and the sample results to complete the test of the hypotheses. Use a 5% significance level. Assume the results come from a random sample, and if the sample size is small, assume the underlying distribution is relatively normal.Test H0 : μ=15 vs Ha : μ>15 using the sample results x¯=17.2, s=6.4, with n=40. (a) Give the test statistic and the p-value.Round your answer for the test statistic to two decimal places and your answer for the p-value to three decimal places.test statistic = p-value =
Test the claim about the population variance o² at the level of significance a. Assume the population is normally distributed. Claim: o² ≤17.1; x = 0.01 Sample statistics: s² = 25.13, n = 41 Write the null and alternative hypotheses. Ho: H₂:0²
We want to conduct a hypothesis test of the claim that the population mean time it takes drivers to react following the application of brakes by the driver in front of them is more than 2.3 seconds. So, we choose a random sample of reaction time measurements. The sample has a mean of 2.2 seconds and a standard deviation of 0.5 seconds. For each of the following sampling scenarios, choose an appropriate test statistic for our hypothesis test on the population mean. Then calculate that statistic. Round your answers to two decimal places. (a) The sample has size 12, and it is from a normally distributed population with an unknown standard deviation. O z = 0 %3D Ot= It is unclear which test statistic to use. (b) The sample has size 17, and it is from a population with a distribution about which we know very little. O z t =|| D- It is unclear which test statistic to use.

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