a) A random sample of eight observations was taken from a normal population. The sample mean and standard deviation are 75 and 50, respectively. Can we infer at the 10% significance level that the population mean is less than 100?b) Repeat part (a) assuming that you know that the population standard deviation is 50.c) Review parts (a) and (b). Explain why the test statistics differ.

A sample of 39 observations is selected from a normal population. The sample mean is 43, and the population standard deviation is 6. Conduct the following test of hypothesis using the 0.02 significance level.

<p>a) To infer if the population mean is less than 100 at the 10% significance level, we can perform a one-sample t-test.</p> <p>Step 1: State the null hypothesis (H0) and the alternative hypothesis (H1). H0: μ = 100 (The population mean is equal to 100) H1: μ &lt; 100 (The population mean is less than 100)</p> <p>Step 2: Calculate the test statistic. The formula for the t statistic is:</p> <p>t = (x̄ - μ) / (s/√n)</p> <p>where x̄ is the sample mean, μ is the population mean, s is the sample standard deviation, and n is the sample size.</p> <p>t = (75 - 100) / (50/√8) = -1.414</p> <p>Step 3: Determine the critical value from the t-distribution table. For a one-tailed test at the 10% significance level with 7 degrees of freedom (n-1), the critical value is -1.415.</p> <p>Step 4: Compare the test statistic with the critical value. If the test statistic is less than the critical value, we reject the null hypothesis. In this case, -1.414 is greater than -1.415, so we do not reject the null hypothesis. Therefore, we cannot infer at the 10% significance level that the population mean is less than 100.</p> <p>b) If we know that the population standard deviation is 50, we can perform a z-test instead of a t-test.</p> <p>Step 1: The null and alternative hypotheses are the same as in part (a).</p> <p>Step 2: Calculate the z statistic using the formula:</p> <p>z = (x̄ - μ) / (σ/√n)</p> <p>where σ is the population standard deviation.</p> <p>z = (75 - 100) / (50/√8) = -1.414</p> <p>Step 3: Determine the critical value from the z-distribution table. For a one-tailed test at the 10% significance level, the critical value is -1.28.</p> <p>Step 4: Compare the z statistic with the critical value. In this case, -1.414 is less than -1.28, so we reject the null hypothesis. Therefore, we can infer at the 10% significance level that the population mean is less than 100.</p> <p>c) The test statistics differ because a t-test is used when the population standard deviation is unknown and is estimated from the sample, while a z-test is used when the population standard deviation is known. The t-distribution is wider and has heavier tails than the z-distribution, which leads to a larger critical value and a higher chance of not rejecting the null hypothesis.</p> ####

The nine item of samples had following value 45,47,50,52,48, 47,49,53 \& 51 .Does the mean of nine item differ significantly from population mean of 47.5 ?

Suppose we take repeated random samples of size 20 from a population with a mean of 60 and a standard deviation of 8. Which of the following statements is true about the sampling distribution of the sample mean (x̄)? Check all that apply. The distribution is normal regardless of the shape of the population distribution, because the sample size is large enough. The distribution will be normal as long as the population distribution is normal. The distribution's mean is the same as the population mean 60. The distribution's standard deviation is larger than the population standard deviation of 8.

You draw 10 samples (n=100) out of a population of 1000 people. If the sample mean of one of the samples is very different from the population mean, does this means this sample should not be analyzed?Question 1Answera.Trueb.FalseClear my choiceQuestion 2Not yet answeredMarked out of 1.00Flag questionTipsQuestion textFrom the distribution of a random variable X a random sample of size n is drawn. What is the distribution of ?Answer:A. normally distributedB. C.  a and bQuestion 2Answera.Ab.Bc.CClear my choiceQuestion 3Not yet answeredMarked out of 1.00Flag questionTipsQuestion textWhen you can apply the Central Limit Theorem, what does it tell us?Question 3Answera.population distribution is normally distributed.b.sampling distribution is normally distributed.c.population standard deviation is known.Clear my choiceQuestion 4Not yet answeredMarked out of 1.00Flag questionTipsQuestion textFrom the distribution of a random variable X a random sample of size n is drawn. What does x̄ represent?Question 4Answera.meanb.sample meanc.standard deviationClear my choiceQuestion 5Not yet answeredMarked out of 1.00Flag questionTipsQuestion textWhat does the law of large number say?Question 5Answera.as the sample size gets larger, the sample mean gets closer and closer to the population mean.b.as the sample size gets larger, the distribution of the sample means approaches a normal distributionc.a &b