In the 1970s a study was conducted in Philadelphia in which 500 cases were randomly assigned to treatments for the common cold: 250 subjects received the medication and 250 received a placebo. A total of 383 patients improved within 24 hours. Of those who received the medication 241 improved within 24 hours and of those who received the placebo 142 improved within 24 hours. A test of significance was conducted on the following hypotheses. H_o: The rates for the two treatments are equal. H_a: The treatment of medication has a higher improvement rate. This test resulted in a p-value of 0.0761. a.) Interpret what this p-value measures in the context of this study. b.) Based on this p-value and study design, what conclusion should be drawn in the context of this study? Use a significance

In the 1970s a study was conducted in Philadelphia in which 500 cases were randomly assigned to treatments for the common cold: 250 subjects received the medication and 250 received a placebo. A total of 383 patients improved within 24 hours. Of those who received the medication 241 improved within 24 hours and of those who received the placebo 142 improved within 24 hours. A test of significance was conducted on the following hypotheses. H_o: The rates for the two treatments are equal. H_a: The treatment of medication has a higher improvement rate. This test resulted in a p-value of 0.0761. a.) Interpret what this p-value measures in the context of this study. b.) Based on this p-value and study design, what conclusion should be drawn in the context of this study? Use a significance

Question
Study design
asked 2021-03-02
In the 1970s a study was conducted in Philadelphia in which 500 cases were randomly assigned to treatments for the common cold: 250 subjects received the medication and 250 received a placebo. A total of 383 patients improved within 24 hours. Of those who received the medication 241 improved within 24 hours and of those who received the placebo 142 improved within 24 hours. A test of significance was conducted on the following hypotheses.
\(\displaystyle{H}_{{o}}\): The rates for the two treatments are equal.
\(\displaystyle{H}_{{a}}\): The treatment of medication has a higher improvement rate.
This test resulted in a p-value of 0.0761.
a.) Interpret what this p-value measures in the context of this study.
b.) Based on this p-value and study design, what conclusion should be drawn in the context of this study? Use a significance level of \(\displaystyle\alpha={0.05}\).
c.) Based on your conclusion in part (b), which type of error, Type I or Type II, could have been made? What is one potential consequence of this error?

Answers (1)

2021-03-03
Step 1
a)
The p-value is 0.0761.
Thus, the probability of obtaining a test statistic value at least as extreme as the observed value, when the rates of two treatments are equal, is 0.0761.
Step 2
b)
Null hypothesis:
\(\displaystyle{H}_{{0}}\): The rates of two treatments are equal.
Alternative hypothesis:
\(\displaystyle{H}_{{a}}\): The treatment of medication has a higher improvement rate.
Decision rule:
If p-value is less than or equal to level of significance, then reject the null hypothesis. Otherwise, fail to reject the null hypothesis.
As p-value of 0.0761 is greater than 0.05, fail to reject the null hypothesis.
Hence, there is not enough evidence to claim that the treatment of medication has a higher improvement rate, at 0.05 level of significance.
c)
Type I error:
The type I error is the rejection of the null hypothesis when it is actually true.
Type II error:
The type II error is the failure of rejection of the null hypothesis when alternative hypothesis is true.
In this case, there is not enough evidence to claim that the treatment of medication has a higher improvement rate. However, there might be a chance that actually the treatment of medication has a higher improvement rate.
Hence there is a chance of type II error.
When type II error occurs in this case, one would believe that medication does not have a higher improvement rate, and would see no need of applying medication, when in reality medication leads to improved rate.
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The table below shows the number of people for three different race groups who were shot by police that were either armed or unarmed. These values are very close to the exact numbers. They have been changed slightly for each student to get a unique problem.
Suspect was Armed:
Black - 543
White - 1176
Hispanic - 378
Total - 2097
Suspect was unarmed:
Black - 60
White - 67
Hispanic - 38
Total - 165
Total:
Black - 603
White - 1243
Hispanic - 416
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Give your answer as a decimal to at least three decimal places.
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This just means that the percentage of times that both things happen equals the individual percentages multiplied together (Only if they are Independent of each other).
Therefore, if a person's race is independent of whether they were killed being unarmed then the percentage of black people that are killed while being unarmed should equal the percentage of blacks times the percentage of Unarmed. Let's check this. Multiply your answer to part a (percentage of blacks) by your answer to part b (percentage of unarmed).
Remember, the previous answer is only correct if the variables are Independent.
d) Now let's get the real percent that are Black and Unarmed by using the table?
If answer c is "significantly different" than answer d, then that means that there could be a different percentage of unarmed people being shot based on race. We will check this out later in the course.
Let's compare the percentage of unarmed shot for each race.
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f) What percent are Hispanic and Unarmed?
If you compare answers d, e and f it shows the highest percentage of unarmed people being shot is most likely white.
Why is that?
This is because there are more white people in the United States than any other race and therefore there are likely to be more white people in the table. Since there are more white people in the table, there most likely would be more white and unarmed people shot by police than any other race. This pulls the percentage of white and unarmed up. In addition, there most likely would be more white and armed shot by police. All the percentages for white people would be higher, because there are more white people. For example, the table contains very few Hispanic people, and the percentage of people in the table that were Hispanic and unarmed is the lowest percentage.
Think of it this way. If you went to a college that was 90% female and 10% male, then females would most likely have the highest percentage of A grades. They would also most likely have the highest percentage of B, C, D and F grades
The correct way to compare is "conditional probability". Conditional probability is getting the probability of something happening, given we are dealing with just the people in a particular group.
g) What percent of blacks shot and killed by police were unarmed?
h) What percent of whites shot and killed by police were unarmed?
i) What percent of Hispanics shot and killed by police were unarmed?
You can see by the answers to part g and h, that the percentage of blacks that were unarmed and killed by police is approximately twice that of whites that were unarmed and killed by police.
j) Why do you believe this is happening?
Do a search on the internet for reasons why blacks are more likely to be killed by police. Read a few articles on the topic. Write your response using the articles as references. Give the websites used in your response. Your answer should be several sentences long with at least one website listed. This part of this problem will be graded after the due date.
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(b) Find the probability that arandomly selected sample of Pima clay loam will have bulk densityless than \(\displaystyle{0.9}\frac{{g}}{{c}}{m}^{{3}}\).
(c) Would you be surprised if a randomly selected sample of this type of soil has a bulkdensity in excess of \(\displaystyle{2.0}\frac{{g}}{{c}}{m}^{{3}}\)? Explain, based on theprobability of this occurring.
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