From past experiences it is known that a certain used car salesperson has a 20% success rate with their clients.  The salesperson receives some training to improve their sales strategies.  Let X be the random variable denoting the number of used car sales by the salesperson after they have received their training, and assume that X follows a Binomial distribution. Over the next 20 clients, it is observed that the salesperson sells 8 used cars.  Carry out a hypothesis test to decide whether there is significant evidence to conclude that the training has increased the success probability of car sales.  Use .Do not forget to define a parameter, state your hypotheses, calculate a p-value, make a decision and a conclusion.

Step 1: Define the parameter The parameter of interest is p, the probability of a successful sale after the training.

Step 2: State the hypotheses The null hypothesis (H0) is that the training has not increased the success probability, i.e., p = 0.20. The alternative hypothesis (H1) is that the training has increased the success probability, i.e., p > 0.20.

Step 3: Calculate the test statistic and p-value The test statistic for a binomial test is calculated as (X - np0) / sqrt(np0(1-p0)), where X is the number of successes, n is the number of trials, and p0 is the hypothesized probability of success under the null hypothesis. Here, X = 8, n = 20, and p0 = 0.20. So, the test statistic is (8 - 200.20) / sqrt(200.20*(1-0.20)) = 2.

The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the observed test statistic, under the null hypothesis. For a one-sided test like this, the p-value is calculated as P(Z > 2), where Z is a standard normal random variable. This can be found using a standard normal table or a statistical software.

Step 4: Make a decision If the p-value is less than the significance level (0.05), we reject the null hypothesis. If the p-value is greater than the significance level, we do not reject the null hypothesis.

Step 5: Conclusion Based on the decision in step 4, we conclude whether there is significant evidence to suggest that the training has increased the success probability of car sales. If we reject the null hypothesis, we conclude that there is significant evidence to suggest that the training has increased the success probability. If we do not reject the null hypothesis, we conclude that there is not significant evidence to suggest that the training has increased the success probability.