## One-Tailed Tests | Statistics 1 | Maths A Level & As Level

alarm_on23-Mar-2022

Welcome to the session on hypothesis, testing one-tailed tests, the objectives for this session are to carry out and interpret a one-tailed test for the proportion of the binomial distribution, we're going to begin this session by highlighting key points on how to carry out a one-tailed hypothesis test and apply these steps to an example. So let's begin in order to carry out a one-tailed hypothesis test first formulate a model for the test statistic next identify null and alternative hypotheses. Then. Calculate probability of the test statistic taking the observed value or higher or lower. Assuming that the null hypothesis is true. Then compare it to the significance level and finally write the conclusion in the context of the question. So let's.

Now take a look at an example, the success rate of the standard treatment for patients suffering from a certain disease is 37 percent. A doctor has produced a new drug, which has been successful with 13 out of 25 patients, the doctor claims that the new drug. Represents an improvement in the standard of the treatment test at the five percent significance level, the claim made by the doctor. So as mentioned earlier, we are going to apply the procedure to this example. So let's solve this. Let x be the number of patients in the trial for whom the drug has been successful. And let p be the probability of success for each patient.

Therefore, we have that x is distributed binomial with n equal to 25, because there are 25 patients in the trial and p equal to the. Probability so this is our model for the test statistic, and we've therefore carried out the first step now, let's, identify the null and alternative hypotheses so h naught the null hypothesis is such that p equals 0.37. This represents the success rate of the standard treatment for patients suffering from a certain disease. Currently, the alternative hypothesis h1 is p greater than 0.37.

The doctor has claimed that the new drug represents an improvement in the standard of the treatment. So because. We're, requiring a test of the whether it's improved or not we're looking at a one-tailed test in one direction.

We are now going to calculate the probability of the test statistic taking the observed value or higher or lower than it. And of course, in our example, we're testing for higher than and assuming that the null hypothesis is true. So we assume h naught is true, and therefore x is distributed binomial with n, equal to 25 and p equal to 0.37 we're asking the question. What is the probability. That x is greater than or equal to thirteen. This is equal to one minus the probability that x is less than or equal to 12. We can either find probability x less than or equal to 12 from our calculator, or from the tables or type in Excel here, we're going to type in Excel equals by norm.

Dist open bracket then plug in 12 here. And here as you can see plug in n equals 25 here and plug in p equals 0.37 here. Once we've typed this into Excel, we obtain 0.9093. And of course, 1 minus 0.9093 is 0.0907.

This. Is equal to 9.07 percent. Finally, we compare this to the significance level and clearly the significance level here we have nine point. Zero, seven percent is greater than five percent. And therefore there is not enough evidence to reject h naught. So finally, we can write the conclusion in the context of the question.

We can conclude, therefore that the new drug is not better than the old one at the 5 significance level. There is another way of attempting. This example, or this type of example. So in order. To carry out a one-tailed test find the critical region and see if the observed value of the test statistic lies inside it. So let's solve this same example, using the second method we let x be the number of patients in the trial for whom the drug has been successful.

And let p be the probability of success for each patient. So again, x is distributed binomial with n equal to 25. And p the probability the hypotheses remain the same as in the previous method h naught is such that peak was 0.37 h1 is. Such that p is greater than 0.37 and remember in Excel when we're using the formula to find the smallest r with probability x greater than or equal to r, which is less than alpha for x. Distributed binomial, open bracket n. P, we type equals by not dot, inf open bracket n. Comma, p, comma, one minus alpha then add one so let's apply this formula to our example in Excel. We type equals by not dot in open bracket n, equal to 25 p equals 0.37 and remember because we're testing at the 5 level, this is 0.05.

So.1 minus alpha or 1, minus 0.05 is 0.95. So we type in 0.95 as shown and then close the brackets. This gives us thirteen. So we add one to give r equal to thirteen plus one. And of course, 13 plus 1 gives us 14. Therefore, the critical region is 14 or more. Now since we're, looking at 14 or more being in the critical region.

And since we have 13 out of 25 having a success rate with the new drug, we can conclude that 13 is not in the critical region. Therefore, there is not enough evidence to reject h. Naught. And therefore, the new drug is not better than the old one at the five percent significance level. This brings us to the end of this session.

If you have understood its contents, you should be able to carry out and interpret a one-tailed test for the proportion of the binomial distribution. Thank you very much for listening to understand this topic in greater depth. Do attempt the exercises in your textbook related to this topic.