# Researchers wanted to compare the effectiveness of a water softener used with a filtering process with a water softener used without filtering. Ninety locations were randomly divided into two groups of equal size. Group A locations used a water softener and the filtering process, while group B used only the water softener. At the end of three months, a water sample was tested at each location for its level of softness. (Water softness was measured on a scale of 1 to 5, with 5 being the softest water.) The results were as follows: Group A (water softener and filtering) x_1 = 2.1 s_1 = 0.7 Group B (water softener only) x_2 = 1.7 s_2 = 0.4 Determine, at the 90% confidence level, whether there is a difference between the two types of treatments.

Question
Comparing two groups
Researchers wanted to compare the effectiveness of a water softener used with a filtering process with a water softener used without filtering. Ninety locations were randomly divided into two groups of equal size. Group A locations used a water softener and the filtering process, while group B used only the water softener. At the end of three months, a water sample was tested at each location for its level of softness. (Water softness was measured on a scale of 1 to 5, with 5 being the softest water.) The results were as follows: Group A (water softener and filtering) $$\displaystyle{x}_{{1}}={2.1}$$
$$\displaystyle{s}_{{1}}={0.7}$$ Group B (water softener only) $$\displaystyle{x}_{{2}}={1.7}$$
$$\displaystyle{s}_{{2}}={0.4}$$ Determine, at the 90% confidence level, whether there is a difference between the two types of treatments.

2020-10-24

Group A $$\displaystyle{x}_{{1}}={2.1}$$
$$\displaystyle{s}_{{1}}={0.7}$$
$$\displaystyle{n}_{{1}}={45}$$ Group B $$\displaystyle{x}_{{2}}={1.7}$$
$$\displaystyle{s}_{{2}}={0.4}$$
$$\displaystyle{n}_{{2}}={45}$$ $$\displaystyle{S}.{E}.=\sqrt{{{\mathfrak{{a}}}{a}{c}{\left\lbrace{\left({0.7}\right)}^{{2}}\right\rbrace}{\left\lbrace{45}\right\rbrace}+{\frac{{{\left({0.4}\right)}^{{2}}}}{{{45}}}}}}={0.120}$$
$$\displaystyle{d}={x}_{{1}}-{x}_{{2}}={2.1}-{1.7}={0.4}$$
$$\displaystyle{t}_{{c}}{n}_{{c}}{\left({a}{t}{0.10}{l}{o}{s}{\quad\text{and}\quad}{90}-{2}{d}{f}\right)}={1.65}$$
$$\displaystyle\therefore{C}.{I}.={d}\pm{t}_{{c}}{n}_{{c}}{S}.{E}.$$
$$\displaystyle={0.4}\pm{1.65}{\left({0.120}\right)}$$
$$\displaystyle={0.4}\pm{0.198}$$
$$\displaystyle\therefore{C}.{I}.={\left({0.202},{0.598}\right)}$$

### Relevant Questions

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
Total - 2262
Give your answer as a decimal to at least three decimal places.
a) What percent are Black?
b) What percent are Unarmed?
c) In order for two variables to be Independent of each other, the P $$(A and B) = P(A) \cdot P(B) P(A and B) = P(A) \cdot P(B).$$
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.
e) What percent are White and Unarmed?
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.
Give a full and correct answer Why is it important that a sample be random and representative when conducting hypothesis testing? Representative Sample vs. Random Sample: An Overview Economists and researchers seek to reduce sampling bias to near negligible levels when employing statistical analysis. Three basic characteristics in a sample reduce the chances of sampling bias and allow economists to make more confident inferences about a general population from the results obtained from the sample analysis or study: * Such samples must be representative of the chosen population studied. * They must be randomly chosen, meaning that each member of the larger population has an equal chance of being chosen. * They must be large enough so as not to skew the results. The optimal size of the sample group depends on the precise degree of confidence required for making an inference. Representative sampling and random sampling are two techniques used to help ensure data is free of bias. These sampling techniques are not mutually exclusive and, in fact, they are often used in tandem to reduce the degree of sampling error in an analysis and allow for greater confidence in making statistical inferences from the sample in regard to the larger group. Representative Sample A representative sample is a group or set chosen from a larger statistical population or group of factors or instances that adequately replicates the larger group according to whatever characteristic or quality is under study. A representative sample parallels key variables and characteristics of the large society under examination. Some examples include sex, age, education level, socioeconomic status (SES), or marital status. A larger sample size reduced sampling error and increases the likelihood that the sample accurately reflects the target population. Random Sample A random sample is a group or set chosen from a larger population or group of factors of instances in a random manner that allows for each member of the larger group to have an equal chance of being chosen. A random sample is meant to be an unbiased representation of the larger population. It is considered a fair way to select a sample from a larger population since every member of the population has an equal chance of getting selected. Special Considerations: People collecting samples need to ensure that bias is minimized. Representative sampling is one of the key methods of achieving this because such samples replicate as closely as possible elements of the larger population under study. This alone, however, is not enough to make the sampling bias negligible. Combining the random sampling technique with the representative sampling method reduces bias further because no specific member of the representative population has a greater chance of selection into the sample than any other. Summarize this article in 250 words.
Two different analytical methods were used to determine residual chlorine in sewage effluents. Both methods were used on the same samples, but each came from various locations, with differing amounts of contact time with the effluent. The concentration of Cl in mg//L was determined by the two methods, and the following results were obtained:
Method $$A: 0.39, 0.84, 1.76, 3.35, 4.69, 7.70, 10.52, 10.92$$
Method $$B: 0.36, 1.35, 2.56, 3.92, 5.35, 8.33, 10.70, 10.91$$
(a) What type of t test should be used to compare the two methods and why?
(b) Do the two methods give different results?
(c) Does the conclusion depend on whether the 90%, 95% or 99% confidence levels are used?
Identify the appropriate hypothesis test for each of the following research situations using the options: The null hypothesis, The Test Statistics, The Sample Statistic, The Standard Error, and The Alpha Level.
A researcher conducts a cross-sectional developmental study to determine whether there is a significant difference in vocabulary skills between 8-year-old and 10-year-old children. A researcher determines that 8% of the males enrolled in Introductory Psychology have some form of color blindness, compared to only 2% of the females. Is there a significant relationship between color blindness and gender?
A researcher records the daily sugar consumption and the activity level for each of 20 children enrolled in a summer camp program. The researcher would like to determine whether there is a significant relationship between sugar consumption and activity level.
A researcher would like to determine whether a 4-week therapy program produces significant changes in behavior. A group of 25 participants is measured before therapy, at the end of therapy, and again 3 months after therapy.
A researcher would like to determine whether a new program for teaching mathematics is significantly better than the old program. It is suspected that the new program will be very effective for small-group instruction but probably will not work well with large classes. The research study involves comparing four groups of students: a small class taught by the new method, a large class taught by the new method, a small class taught by the old method, and a large class taught by the old method.
Potential buyers for a new car were randomly divided into two groups. One group was shown the "A" version of an ad for the car, while the other group was shown the "B" version of the ad. All were then tested on their recall of key points made in the ad. The researcher should run a hypothesis test based upon a comparison of means for ?
In another study, a healthcare insurance company took measures of subscribersâ€™ cardiac (heart) health. The people were then provided an app for their phones which provided "nudges" and reminders about heart-healthy behaviors, such as eating more vegetables and less fried or fatty food, taking walks and breaks from sitting too long, and getting enough sleep. After 4 months of having the app, the cardiac health measures were taken again, with the objective of seeing if nudges from the app would result in decreased cardiac risk. The researcher should run a hypothesis test based on a comparison of means for?
The American Journal of Political Science (Apr. 2014) published a study on a woman's impact in mixed-gender deliberating groups. The researchers randomly assigned subjects to one of several 5-member decision-making groups. The groups' gender composition varied as follows: 0 females, 1 female, 2 females, 3 females, 4 females, or 5 females. Each group was the n randomly assigned to utilize one of two types of decision rules: unanimous or majority rule. Ten groups were created for each of the $$\displaystyle{6}\ \times\ {2}={12}$$ combinations of gender composition and decision rule. One variable of interest, measured for each group, was the number of words spoken by women on a certain topic per 1,000 total words spoken during the deliberations. a) Why is this experiment considered a designed study? b) Identify the experimental unit and dependent variable in this study. c) Identify the factors and treatments for this study.
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