Examples of testing the mean and the proportion with single samples Some Hypotheses Tests Examples of testing the mean and the proportion with single samples P-value method These are just some example of hypothesis tests using one sample. Please read the text and be sure you understand the basic concepts of hypothesis testing. This presentation probably will not cover all the material in the text.

In the 1980’s it was generally believed that autism affected about 8% of the nation’s children. Some people believe that the increase in the number of chemicals in the environment has led to an increase in the incidence of autism. A recent study examined 384 children and found that 46 of them showed signs of some form of autism. Is there strong evidence that the level of autism has increased at the 5% significance level? Fail to reject region Read the problem----- First decide what parameter the problem addresses. In this case- we are looking at the population proportion of children with autism. I know that because the problem mentions that it is believed that the percent of children with autism is 8% and, because the problem also reports a ratio when speaking about children with autism A mean is not mentioned. Since the concern is that the old proportion has INCREASED, the alternative hypothesis shows that perhaps the new proportion, p, could be greater than .08. Remember that the null hypothesis will always show equality. If our sampling shows that the percent of cases is well beyond 8%, we will reject the null hypothesis and conclude that there has been an increase. If we find a percent at or less than 8%, then the new findings support the old 8%. This means that our test is called a one-tailed test and our focus is in the upper tail of the normal curve. rejection region

Using the calculator Stat…tests…(5)1-PropZ-test..enter

Use the p value to help make the decision. The number here that is going to help us decide if the incidence of autism has increased, is the p=value. A p-value of .002 means that, if the null hypothesis is true, the chance of what we just saw happen, is .2%. So what do you think about that? It tells us that something with a very rare .2% chance of happening, just happened. Well that most likely means that the null hypothesis is not true and that we should reject it and go with the idea behind the alternative hypothesis.

Decision and conclusion Decision: Reject the null hypothesis Conclusion: There has been an increase in the number of children with autism. So, because the p value was .002, very small, we will decide to reject the null hypothesis which states that the percent of children with autism is 8%.We can then conclude that the percent of children with autism has increased from the old 8%.

The heart rate of a healthy lion is approximately normally distributed with a mean of 40 beats per minute. A heart rate that is too slow or too fast can indicate a health problem. A vet has observed the following beats of a young lion's heart who is recovering from surgery. Do these data indicate that the lion may have a health problem? 30 37 43 38 35 36 rejection region Fail to reject region Let's try another one Read problem--- Now this one is a little different than the first one. This problem is addressing the population mean. Therefore, we will be doing a test on mu. Another difference is that we actually have the data from this study. The problem points out that it is important that the heart beat be around 40 beat per minutes, no more, no less. Otherwise there is a health problem. So--- this is a two tailed test. If our data places us in the right tail or the left tail, that indicates that the heart beat is too far away fro the mean and conditions are not good for the lion.

Use the calculator Stat…edit- enter the data Stat…tests…T-test.. 30 37 43 38 35 36 Stat…tests…T-test.. To help us do the calculations, you first need to place the data in L1 of the calculator. The follow the keystrokes and do a T-test on the data. If you had places the actual frequencies of each piece of data in list 2, we would indicate that by placing L2 at the frequency section of the screen. However, whenever you merely punch each piece in separately, the frequency of each is simply 1.

Use the p-value to help you decide Once again, look at the p-value. This particular one means that there is a 9% almost 10% chance that what just occurred, will happen again. In other words a mean of 36.5 beats will occur at random 10 out of 100 times or 1 out of 10 times. To most people, a 10% chance of something happening isn’t all that uncommon or rare.

Decide and conclude Decision- do not reject the null hypothesis Conclusion- The lion probably does not have a heart condition since the data show his heart beat is close enough to 40 beats. So, based on the p value, we will decide to not reject the null hypothesis and conclude that the lion’s heart beat is at 40 beats and he is probably okay.

A manufacturer claims that the average life span of its washing machines is at least 4 years. It is assumed that the distribution of the life spans is approximately normal. A random sample of 32 of these machines had an average life span of 3.6 years and a standard deviation of 1.5 years. Is there statistical evidence to dispute the manufacturer's claim? Fail to reject region Read problem Now this one is a little different also. Basically we are testing the life of the washing machines. If they live substantially less than 4 years, then the manufacturer is incorrect and his claim would be false. This is a one tail to the left test. Remember use the sign of the alternative to help you decide which tail is the rejection area. rejection region

Use the calculator Stat…tests…T-test

Use the p-value to help you decide With a p-value of 7%, we find that this is not a rare occurance.

Decide and conclude Decision- fail to reject the null hypothesis Conclusion- the manufacturer is correct in his claim The large p-value will cause us to not reject the null hypothesis and conclude that the manufacturer is coorect in saying that his machines last at least 4 years.

Now, you try some! Now you are armed with some of the skills involved in setting up a hypothesis test. Take a look at the assigned homework problems and try some on your own.