Assume that the cure rate for a certain drug is 20% and that 15 patients are tested with this drug. Find the following:
a) Mean
b) Standard Deviation
c) P (exactly 4 cures)
d) P (exactly 4 cures)
A snack company distributes chips in bags labeled 6 ounces. The local Bureau of Weights and Measures randomly selects 50 bags of chips and obtains a sample mean of 5.9 ounces. Assuming that the standard deviation is known to be 0.8, test the claim at the .05 significance level that the bags contain less than 6 ounces. Identify the following:
a) The Null Hypothesis
b) Critical Value
c) Test Statistics
d) Your Conclusion regarding the Null Hypothesis
e) Your Statement regarding the claim.
To answer the first set of questions:
a) Finding the mean: The cure rate for a certain drug is given as 20%, which can be written as 0.2. To find the mean, you multiply the cure rate by the number of patients tested. In this case, the mean would be 0.2 * 15 = 3.
b) Finding the standard deviation: The standard deviation is a measure of the variation or spread of data. To find the standard deviation, you need to know the variance, which is the square of the standard deviation. Since the variance is not given in this case, we cannot calculate the standard deviation.
c) P (exactly 4 cures): P (exactly 4 cures) refers to the probability of exactly 4 out of 15 patients being cured. To calculate this probability, you can use the binomial probability formula. The formula would be: P(X=k) = (nCk) * p^k * (1-p)^(n-k), where n is the number of trials (15 in this case), k is the number of successful outcomes (4), p is the probability of success (0.2), and (nCk) represents the combination notation. Plugging in the values, you can calculate P (exactly 4 cures).
d) P (at least 4 cures): To find the probability of at least 4 cures, you need to calculate the probabilities for exactly 4, exactly 5, exactly 6, and so on, until 15 cures. Then, you add up all these individual probabilities to find the probability of at least 4 cures.
Now, moving on to the second set of questions:
a) The Null Hypothesis: The null hypothesis is a statement that assumes there is no significant difference or effect. In this case, the null hypothesis would be that the bags of chips contain 6 ounces (the claimed amount).
b) Critical Value: The critical value is a threshold value that is used to determine whether the null hypothesis should be rejected or not. The critical value depends on the significance level and the degrees of freedom. In this case, a significance level of 0.05 is given, so you would need to look up the critical value for a t-test with 49 degrees of freedom.
c) Test Statistics: The test statistic is a value calculated from sample data that is used to assess the likelihood of the null hypothesis being true. In this scenario, the test statistic would be calculated using the formula: (sample mean - hypothesized mean) / (standard deviation / sqrt(n)). Plugging in the given values, you can calculate the test statistic.
d) Your Conclusion regarding the Null Hypothesis: To draw a conclusion regarding the null hypothesis, you compare the test statistic with the critical value. If the test statistic falls within the rejection region (determined by the critical value), you reject the null hypothesis. Otherwise, you fail to reject the null hypothesis.
e) Your Statement regarding the claim: Based on your conclusion, you can make a statement regarding the claim. In this case, the claim is that the bags contain less than 6 ounces. If you reject the null hypothesis, you would support the claim. If you fail to reject the null hypothesis, you would not have enough evidence to support the claim.