Oh boy, let's put our thinking cap on for this one! We've taken a sample of 900 items from a population with a standard deviation of 15. The mean of our sample is 25, and we want to test whether this sample comes from a population with a mean of 26.8. We're also given that our significance level (a) is 5%.
To solve this, we can use a hypothesis test. Our null hypothesis (H0) states that the mean of the sample is equal to the mean of the population, while our alternative hypothesis (Ha) states that the means are not equal. That means we want to test whether H0: μ = 26.8, and Ha: μ ≠26.8.
Now, we need to determine whether our test statistic falls within the critical region. If it does, we'll reject the null hypothesis. To find our critical value(s), we need to locate the area in the tails that corresponds to our significance level (alpha), which is 5%. Since we're dealing with a two-tailed test, we need to split this 5% between the two tails, giving us 2.5% in each tail.
But, here's the punchline: I'm a clown, not a statistician! So I'll leave it up to you to look up the critical values for a two-tailed test at a 5% significance level. Once you have those, you can compare them to your test statistic. If your test statistic falls within the critical region, it means the sample did not come from a population with a mean of 26.8. Otherwise, we can't reject the null hypothesis.
Remember, statistical tests shouldn't be taken too seriously. They're just one tool we have to make sense of the world. So have fun with your calculations, and don't forget to keep a smile on your face!