When we execute not know the population standard deviation sigma, σ, and the sample size, n, is much less than 30, we use the t-test to advice a claim. Think about the complying with problem:
A scientist think the mean waste recycle by adult in the united state is now an ext than one lb per human per day. In a random sample of 12 us adults, the typical waste recycled per day per human being is 1.9 pounds with a traditional deviation that 0.3 pounds. At a 10% definition level, go the sample data assistance the claim?
In statistics, the is constantly a an excellent idea to lay out the situation described in the problem:
The sample typical is much to the best of the assumed population mean, µ = 1.
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To resolve this problem, we start by stating the null and alternative hypotheses:The insurance claim includes the phrase “more than” which shows the math operator is > .Because the null should be a kind of equality < ≤, =, or ≥ >, the case is the alternative. Ho: µ ≤ 1; Ha: µ > 1.
We deserve to perform the hypothesis test one of two ways:Find the an important value of t and determine if the test statistic is in the rejection region.Calculate the p-value for the test statistic and compare it come alpha.
Rejection an ar approach.The mathematics operator in the alternative, >, points to the right, therefore this is a right-tail test.To discover the an essential value of t, we require the levels of freedom.N = 12df = n-1 = 11Using Excel,Using the StatCrunch command sequence, Stat > Calculators > T. Get in the df and also alpha. Select the “≥” operator to gain the ideal tail. Click Compute. I favor StatCrunch for this due to the fact that it always creates the map out which helps avoid “dumb” tail errors.
The an essential value of t is 1.36 and also the rejection region is t > 1.36, the area in red.Now we need to find the check statistic.
Although I have the right to do this utilizing the Compute function in StatCrunch, I favor to usage Excel because that this since I can save the record and reuse on similar problems:
Since t = 10.39 drops in the rejection region to the appropriate of t-critical = 1.36, we refuse the null hypothesis.
Now let’s usage the p-value approach.First, using Excel:Now using StatCrunch: usage the command sequence Stat > T Stats > One Sample > with Summary. Enter the sample mean, sample standard deviation, and n. Note: execute not transform the sample s come the traditional error (sigma below x-bar) as we did utilizing Excel since StatCrunch has that step built-in.
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StatCrunch gives has the test statistic, 10.392, in the output in addition to the p-value
Getting the trust interval using Excel is not difficult though you have to recall we use the critical value the t because that alpha/2, which is the two-tail critical value even though we were to run a one-tail theory test. That trips increase a the majority of students who just use the t-critical for the rejection area.