{"id":5065,"date":"2022-08-17T20:41:25","date_gmt":"2022-08-17T20:41:25","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/?post_type=chapter&p=5065"},"modified":"2022-08-17T20:43:18","modified_gmt":"2022-08-17T20:43:18","slug":"13d-preview","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/chapter\/13d-preview\/","title":{"raw":"13D Preview","rendered":"13D Preview"},"content":{"raw":"Preparing for the next class\r\n\r\nIn the next in-class activity, you will need to be able to calculate the mean of a\u00a0 difference and identify the differences between independent and dependent samples.\r\n\r\nDependent Samples vs. Independent Samples\r\n\r\nPreviously, you learned how to create confidence intervals and conduct hypothesis tests\u00a0 with a single variable. You also learned how to compare means or proportions from two\u00a0 samples. Some statistical studies use samples from more than one population. In order\u00a0 to compare the difference between two populations, it is important to identify if the\u00a0 samples are dependent (paired) or independent. Dependent and independent sample hypothesis tests are used to answer questions about the difference between two population means.\r\n\r\nFor dependent (paired) samples, the same variable is recorded for each sample, and\u00a0 there is a logical way to pair the observations from one sample with the observations in\u00a0 the other sample. In contrast, when samples are independently selected, the same\u00a0 variable is measured for both samples, but there is no logical way to pair an observation\u00a0 from one sample with a particular observation from the other sample.\r\n\r\nFor an example of paired samples, consider an investigation on the effectiveness of\u00a0 hypnosis in reducing pain. The variable could be the pain level of a patient, and it could\u00a0 be measured \u201cbefore\u201d hypnosis and then again \u201cafter\u201d hypnosis for the same patient.\u00a0 This would result in two samples, one \u201cbefore\u201d pain measurement and one \u201cafter\u201d pain\u00a0 measurement, and there would be a logical pairing of the \u201cbefore\u201d measurement with\u00a0 the \u201cafter\u201d measurement for the same person. This form of pairing, often referred to as\u00a0 \u201cpre\/post,\u201d is not the only situation where paired samples can be used. Other cases\u00a0 involve using \u201cnatural pairs,\u201d such as twins, siblings, or couples. In either case, it is not\u00a0 reasonable that the measurement from one sample is not related to the measurement in\u00a0 the second sample.\r\n\r\nQuestions 1\u20137: Use the previous information to determine if the following situations\u00a0 would result in dependent or independent samples.\r\n
\r\n

Question 1<\/h3>\r\n1) A company that creates fishing accessories is researching two of their most popular\u00a0 fishing rods. The company collects a random sample of the number of sales for each\u00a0 fishing rod from 100 of their stores.\r\n\r\n<\/div>\r\n
\r\n

Question 2<\/h3>\r\n2) The North Carolina Zoo is researching whether their animals are more active in the\u00a0 morning or in the evening. An employee at the zoo visits each habitat in the zoo and\u00a0 collects information for the study. The employee counts how many of each species\u00a0 is visible in the morning and then visits a second time to count how many of each\u00a0 species is visible during the evening.\r\n\r\n<\/div>\r\n
\r\n

Question 3<\/h3>\r\n3) A company that creates blood pressure medicine is researching the effectiveness of\u00a0 their new blood pressure medicine. The company conducts a study in which\u00a0 volunteers are randomly assigned to two groups. One group is given the new\u00a0 medication and the other group continues to take their current blood pressure\u00a0 medicine.\r\n\r\n<\/div>\r\n
\r\n

Question 4<\/h3>\r\n4) The same company that creates blood pressure medicine is still researching the\u00a0 effectiveness of their new blood pressure medicine. The company conducts a\u00a0 second study in which volunteers are all given the new medication. The blood\u00a0pressure of each patient is measured before the study begins. The patients are all\u00a0 given the new medication for six weeks. The blood pressure of each patient is\u00a0 measured after the six-week period.\r\n\r\n<\/div>\r\n
\r\n

Question 5<\/h3>\r\n5) A psychologist wants to know if children\u2019s levels of anxiety are different if their\u00a0 parents are divorced. The psychologist decides to study 100 children from divorced\u00a0 parents and 100 children from non-divorced parents.\r\n\r\n<\/div>\r\n
\r\n

Question 6<\/h3>\r\n6) The quality control manager at a manufacturing plant is investigating the production\u00a0 rate of two machines that were built with the same materials and the same design\u00a0 but were manufactured at two different plants.\r\n\r\n<\/div>\r\n
\r\n

Question 7<\/h3>\r\n7) A statistics teacher wants to know if a curriculum is effective. The teacher conducts\u00a0 a pre-test, implements the curriculum, and then conducts a post-test on the same\u00a0 group of students. The scores on the pre-tests and post-tests are used to compare\u00a0 the difference in understanding of statistics before and after students completed the\u00a0 curriculum.\r\n\r\n<\/div>\r\n
\r\n

Question 8<\/h3>\r\n8) Suppose you want to study the effectiveness of a diet. Suppose that eight people\u00a0 were randomly selected to participate in your study. The weight (lb) of each of the\u00a0 eight participants is recorded before and after the diet in the following table. You\u00a0 know from past studies that body weight is approximately normally distributed.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Patient<\/td>\r\nBefore<\/td>\r\nAfter<\/td>\r\n<\/tr>\r\n
1<\/td>\r\n150<\/td>\r\n146<\/td>\r\n<\/tr>\r\n
2<\/td>\r\n160<\/td>\r\n159<\/td>\r\n<\/tr>\r\n
3<\/td>\r\n200<\/td>\r\n200<\/td>\r\n<\/tr>\r\n
4<\/td>\r\n178<\/td>\r\n174<\/td>\r\n<\/tr>\r\n
5<\/td>\r\n190<\/td>\r\n189<\/td>\r\n<\/tr>\r\n
6<\/td>\r\n167<\/td>\r\n160<\/td>\r\n<\/tr>\r\n
7<\/td>\r\n151<\/td>\r\n148<\/td>\r\n<\/tr>\r\n
8<\/td>\r\n210<\/td>\r\n198<\/td>\r\n<\/tr>\r\n
Mean [latex](\\mu)[\/latex]<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

a) What is the average weight before and after the diet? Fill in your answers in the table.<\/p>\r\nb) On average, how many pounds did the participants lose? In other words,\u00a0 what is the estimated difference between the mean weight before and after the diet?\r\n\r\nc) Are the two samples independent or dependent?\r\n\r\n<\/div>\r\n

\r\n

Question 9<\/h3>\r\n9) Consider the previous example using this new table:\r\n
\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Patient<\/td>\r\nBefore<\/td>\r\nAfter<\/td>\r\nDifference [latex](d)[\/latex]<\/td>\r\n<\/tr>\r\n
1<\/td>\r\n150<\/td>\r\n146<\/td>\r\n146\u2013150 = \u22124<\/td>\r\n<\/tr>\r\n
2<\/td>\r\n160<\/td>\r\n159<\/td>\r\n159\u2013160 = \u22121<\/td>\r\n<\/tr>\r\n
3<\/td>\r\n200<\/td>\r\n200<\/td>\r\n<\/td>\r\n<\/tr>\r\n
4<\/td>\r\n178<\/td>\r\n174<\/td>\r\n<\/td>\r\n<\/tr>\r\n
5<\/td>\r\n190<\/td>\r\n189<\/td>\r\n<\/td>\r\n<\/tr>\r\n
6<\/td>\r\n167<\/td>\r\n160<\/td>\r\n<\/td>\r\n<\/tr>\r\n
7<\/td>\r\n151<\/td>\r\n148<\/td>\r\n<\/td>\r\n<\/tr>\r\n
8<\/td>\r\n210<\/td>\r\n198<\/td>\r\n<\/td>\r\n<\/tr>\r\n
Mean [latex](\\mu)[\/latex]<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\na) How much weight did each individual lose? Complete the table by finding the\u00a0 difference in each participant\u2019s weight (after\u2212before).\r\n\r\nb) Consider ONLY the difference variable. What is the average weight loss for\u00a0 the eight participants?\r\n\r\nc) How does your answer to Question 9, Part B compare to your answer from\u00a0 Question 8, Part B?\r\n\r\n<\/div>\r\n

Comparing Means from Two Dependent (Paired) Samples<\/strong><\/p>\r\nWe will use the individual differences, [latex]d[\/latex], between each pair as our sample. A dependent or paired t-test compares the mean of the differences, [latex]\\mu_{d}[\/latex], to a\u00a0 hypothesized value, which is often 0. Thus, a dependent t-test is the same as a one sample t-test performed on the difference variable, [latex]d[\/latex].\r\n\r\nWhen thinking about the difference variable, we need to use a different calculation for the standard deviation of the estimate. The standard deviation of the difference in the\u00a0 sample means, [latex]\\bar{x}_{1}-\\bar{x}_{2}[\/latex] is NOT the same as the standard deviation of the difference variable, denoted using [latex]s_{d}[\/latex].\r\n\r\nSince a dependent t-test is the same as a one-sample t-test on the mean of the difference variable, the assumptions for a paired t-test are the same as those discussed\u00a0 in In-Class Activity 13.B for a single sample hypothesis test for means.\r\n\r\nConditions for a One-Sample t-Test\r\n

    \r\n \t
  1. The sample is a random sample from the population of interest or it is\u00a0 reasonable to regard the sample as random. It is reasonable to regard the\u00a0 sample as a random sample if it was selected in a way that should result in a\u00a0 sample that is representative of the population.<\/li>\r\n \t
  2. For each population, the distribution of the variable that was measured is\u00a0 approximately normal, or the sample size for the sample from that\u00a0 population is large. Usually, a sample of size 30 or more is considered to be\u00a0 \u201clarge.\u201d If a sample size is less than 30, you should look at a plot of the data\u00a0 from that sample (a dotplot, a boxplot, or, if the sample size isn\u2019t really small, a\u00a0 histogram) to make sure that the distribution looks approximately symmetric\u00a0 and that there are no outliers.<\/li>\r\n<\/ol>\r\nIn summary, where [latex]k[\/latex] is the value of the null hypothesis, we have:\r\n
    \r\n\r\n\r\n\r\n\r\n
    Null Hypothesis for Independent Samples<\/td>\r\nNull Hypothesis for Dependent Samples<\/td>\r\n<\/tr>\r\n
    [latex]H_{0}:\\mu_{1}-\\mu_{2}=k[\/latex]<\/td>\r\n[latex]H_{0}:\\mu_{d}=k[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n
    \r\n\r\n\r\n\r\n\r\n\r\n\r\n
    Alternative Hypothesis for Independent Samples<\/td>\r\nAlternative Hypothesis for Dependent Samples<\/td>\r\n<\/tr>\r\n
    [latex]H_{A}:\\mu_{1}-\\mu_{2}>k[\/latex]<\/td>\r\n[latex]H_{A}:\\mu_{d}>k[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]H_{A}:\\mu_{1}-\\mu_{2}<k[\/latex]<\/td>\r\n[latex]H_{A}:\\mu_{d}<k[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]H_{A}:\\mu_{1}-\\mu_{2}\\neq k[\/latex]<\/td>\r\n[latex]H_{A}:\\mu_{d}\\neq k[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nThe notations for the summary statistics used to compare paired populations\/samples\u00a0 are shown in the following table. We will use[latex]d[\/latex] to represent the difference variable.\r\n
    \r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    Summary Statistics<\/td>\r\nNotation<\/td>\r\n<\/tr>\r\n
    Population Mean of Difference<\/td>\r\n[latex]\\mu_{d}[\/latex]<\/td>\r\n<\/tr>\r\n
    Sample Mean of Difference<\/td>\r\n[latex]\\bar{d}[\/latex]<\/td>\r\n<\/tr>\r\n
    Population Standard Deviation of Difference<\/td>\r\n[latex]\\sigma_{d}[\/latex]<\/td>\r\n<\/tr>\r\n
    Sample Standard Deviation of\u00a0 Difference<\/td>\r\n[latex]s_{d}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n
    \r\n

    Question 10<\/h3>\r\n10) It is a common belief that using higher-octane fuel will improve the gas mileage of a vehicle. In order to test this claim, a mechanic randomly selects 12 customers to\u00a0 participate in a study. The mechanic puts 10 gallons of fuel in each participant\u2019s car\u00a0 and asks participants to circle a racetrack until they run out of gas. Each participant\u00a0 is asked to perform this action two times, once with 87-octane fuel and another time\u00a0 with 92-octane fuel. The differences in miles driven (miles driven with 87-octane fuel and miles driven with 92-octane fuel) are calculated and recorded. The\u00a0 participants do not know which fuels they are using while they are driving around\u00a0 the racetrack.\r\n\r\nWhat are the appropriate null and alternative hypotheses for this scenario?\r\n\r\n<\/div>\r\n ","rendered":"

    Preparing for the next class<\/p>\n

    In the next in-class activity, you will need to be able to calculate the mean of a\u00a0 difference and identify the differences between independent and dependent samples.<\/p>\n

    Dependent Samples vs. Independent Samples<\/p>\n

    Previously, you learned how to create confidence intervals and conduct hypothesis tests\u00a0 with a single variable. You also learned how to compare means or proportions from two\u00a0 samples. Some statistical studies use samples from more than one population. In order\u00a0 to compare the difference between two populations, it is important to identify if the\u00a0 samples are dependent (paired) or independent. Dependent and independent sample hypothesis tests are used to answer questions about the difference between two population means.<\/p>\n

    For dependent (paired) samples, the same variable is recorded for each sample, and\u00a0 there is a logical way to pair the observations from one sample with the observations in\u00a0 the other sample. In contrast, when samples are independently selected, the same\u00a0 variable is measured for both samples, but there is no logical way to pair an observation\u00a0 from one sample with a particular observation from the other sample.<\/p>\n

    For an example of paired samples, consider an investigation on the effectiveness of\u00a0 hypnosis in reducing pain. The variable could be the pain level of a patient, and it could\u00a0 be measured \u201cbefore\u201d hypnosis and then again \u201cafter\u201d hypnosis for the same patient.\u00a0 This would result in two samples, one \u201cbefore\u201d pain measurement and one \u201cafter\u201d pain\u00a0 measurement, and there would be a logical pairing of the \u201cbefore\u201d measurement with\u00a0 the \u201cafter\u201d measurement for the same person. This form of pairing, often referred to as\u00a0 \u201cpre\/post,\u201d is not the only situation where paired samples can be used. Other cases\u00a0 involve using \u201cnatural pairs,\u201d such as twins, siblings, or couples. In either case, it is not\u00a0 reasonable that the measurement from one sample is not related to the measurement in\u00a0 the second sample.<\/p>\n

    Questions 1\u20137: Use the previous information to determine if the following situations\u00a0 would result in dependent or independent samples.<\/p>\n

    \n

    Question 1<\/h3>\n

    1) A company that creates fishing accessories is researching two of their most popular\u00a0 fishing rods. The company collects a random sample of the number of sales for each\u00a0 fishing rod from 100 of their stores.<\/p>\n<\/div>\n

    \n

    Question 2<\/h3>\n

    2) The North Carolina Zoo is researching whether their animals are more active in the\u00a0 morning or in the evening. An employee at the zoo visits each habitat in the zoo and\u00a0 collects information for the study. The employee counts how many of each species\u00a0 is visible in the morning and then visits a second time to count how many of each\u00a0 species is visible during the evening.<\/p>\n<\/div>\n

    \n

    Question 3<\/h3>\n

    3) A company that creates blood pressure medicine is researching the effectiveness of\u00a0 their new blood pressure medicine. The company conducts a study in which\u00a0 volunteers are randomly assigned to two groups. One group is given the new\u00a0 medication and the other group continues to take their current blood pressure\u00a0 medicine.<\/p>\n<\/div>\n

    \n

    Question 4<\/h3>\n

    4) The same company that creates blood pressure medicine is still researching the\u00a0 effectiveness of their new blood pressure medicine. The company conducts a\u00a0 second study in which volunteers are all given the new medication. The blood\u00a0pressure of each patient is measured before the study begins. The patients are all\u00a0 given the new medication for six weeks. The blood pressure of each patient is\u00a0 measured after the six-week period.<\/p>\n<\/div>\n

    \n

    Question 5<\/h3>\n

    5) A psychologist wants to know if children\u2019s levels of anxiety are different if their\u00a0 parents are divorced. The psychologist decides to study 100 children from divorced\u00a0 parents and 100 children from non-divorced parents.<\/p>\n<\/div>\n

    \n

    Question 6<\/h3>\n

    6) The quality control manager at a manufacturing plant is investigating the production\u00a0 rate of two machines that were built with the same materials and the same design\u00a0 but were manufactured at two different plants.<\/p>\n<\/div>\n

    \n

    Question 7<\/h3>\n

    7) A statistics teacher wants to know if a curriculum is effective. The teacher conducts\u00a0 a pre-test, implements the curriculum, and then conducts a post-test on the same\u00a0 group of students. The scores on the pre-tests and post-tests are used to compare\u00a0 the difference in understanding of statistics before and after students completed the\u00a0 curriculum.<\/p>\n<\/div>\n

    \n

    Question 8<\/h3>\n

    8) Suppose you want to study the effectiveness of a diet. Suppose that eight people\u00a0 were randomly selected to participate in your study. The weight (lb) of each of the\u00a0 eight participants is recorded before and after the diet in the following table. You\u00a0 know from past studies that body weight is approximately normally distributed.<\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n
    Patient<\/td>\nBefore<\/td>\nAfter<\/td>\n<\/tr>\n
    1<\/td>\n150<\/td>\n146<\/td>\n<\/tr>\n
    2<\/td>\n160<\/td>\n159<\/td>\n<\/tr>\n
    3<\/td>\n200<\/td>\n200<\/td>\n<\/tr>\n
    4<\/td>\n178<\/td>\n174<\/td>\n<\/tr>\n
    5<\/td>\n190<\/td>\n189<\/td>\n<\/tr>\n
    6<\/td>\n167<\/td>\n160<\/td>\n<\/tr>\n
    7<\/td>\n151<\/td>\n148<\/td>\n<\/tr>\n
    8<\/td>\n210<\/td>\n198<\/td>\n<\/tr>\n
    Mean [latex](\\mu)[\/latex]<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

    a) What is the average weight before and after the diet? Fill in your answers in the table.<\/p>\n

    b) On average, how many pounds did the participants lose? In other words,\u00a0 what is the estimated difference between the mean weight before and after the diet?<\/p>\n

    c) Are the two samples independent or dependent?<\/p>\n<\/div>\n

    \n

    Question 9<\/h3>\n

    9) Consider the previous example using this new table:<\/p>\n

    \n\n\n\n\n\n\n\n\n\n\n\n\n
    Patient<\/td>\nBefore<\/td>\nAfter<\/td>\nDifference [latex](d)[\/latex]<\/td>\n<\/tr>\n
    1<\/td>\n150<\/td>\n146<\/td>\n146\u2013150 = \u22124<\/td>\n<\/tr>\n
    2<\/td>\n160<\/td>\n159<\/td>\n159\u2013160 = \u22121<\/td>\n<\/tr>\n
    3<\/td>\n200<\/td>\n200<\/td>\n<\/td>\n<\/tr>\n
    4<\/td>\n178<\/td>\n174<\/td>\n<\/td>\n<\/tr>\n
    5<\/td>\n190<\/td>\n189<\/td>\n<\/td>\n<\/tr>\n
    6<\/td>\n167<\/td>\n160<\/td>\n<\/td>\n<\/tr>\n
    7<\/td>\n151<\/td>\n148<\/td>\n<\/td>\n<\/tr>\n
    8<\/td>\n210<\/td>\n198<\/td>\n<\/td>\n<\/tr>\n
    Mean [latex](\\mu)[\/latex]<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

    a) How much weight did each individual lose? Complete the table by finding the\u00a0 difference in each participant\u2019s weight (after\u2212before).<\/p>\n

    b) Consider ONLY the difference variable. What is the average weight loss for\u00a0 the eight participants?<\/p>\n

    c) How does your answer to Question 9, Part B compare to your answer from\u00a0 Question 8, Part B?<\/p>\n<\/div>\n

    Comparing Means from Two Dependent (Paired) Samples<\/strong><\/p>\n

    We will use the individual differences, [latex]d[\/latex], between each pair as our sample. A dependent or paired t-test compares the mean of the differences, [latex]\\mu_{d}[\/latex], to a\u00a0 hypothesized value, which is often 0. Thus, a dependent t-test is the same as a one sample t-test performed on the difference variable, [latex]d[\/latex].<\/p>\n

    When thinking about the difference variable, we need to use a different calculation for the standard deviation of the estimate. The standard deviation of the difference in the\u00a0 sample means, [latex]\\bar{x}_{1}-\\bar{x}_{2}[\/latex] is NOT the same as the standard deviation of the difference variable, denoted using [latex]s_{d}[\/latex].<\/p>\n

    Since a dependent t-test is the same as a one-sample t-test on the mean of the difference variable, the assumptions for a paired t-test are the same as those discussed\u00a0 in In-Class Activity 13.B for a single sample hypothesis test for means.<\/p>\n

    Conditions for a One-Sample t-Test<\/p>\n

      \n
    1. The sample is a random sample from the population of interest or it is\u00a0 reasonable to regard the sample as random. It is reasonable to regard the\u00a0 sample as a random sample if it was selected in a way that should result in a\u00a0 sample that is representative of the population.<\/li>\n
    2. For each population, the distribution of the variable that was measured is\u00a0 approximately normal, or the sample size for the sample from that\u00a0 population is large. Usually, a sample of size 30 or more is considered to be\u00a0 \u201clarge.\u201d If a sample size is less than 30, you should look at a plot of the data\u00a0 from that sample (a dotplot, a boxplot, or, if the sample size isn\u2019t really small, a\u00a0 histogram) to make sure that the distribution looks approximately symmetric\u00a0 and that there are no outliers.<\/li>\n<\/ol>\n

      In summary, where [latex]k[\/latex] is the value of the null hypothesis, we have:<\/p>\n

      \n\n\n\n\n
      Null Hypothesis for Independent Samples<\/td>\nNull Hypothesis for Dependent Samples<\/td>\n<\/tr>\n
      [latex]H_{0}:\\mu_{1}-\\mu_{2}=k[\/latex]<\/td>\n[latex]H_{0}:\\mu_{d}=k[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n
      \n\n\n\n\n\n\n
      Alternative Hypothesis for Independent Samples<\/td>\nAlternative Hypothesis for Dependent Samples<\/td>\n<\/tr>\n
      [latex]H_{A}:\\mu_{1}-\\mu_{2}>k[\/latex]<\/td>\n[latex]H_{A}:\\mu_{d}>k[\/latex]<\/td>\n<\/tr>\n
      [latex]H_{A}:\\mu_{1}-\\mu_{2}\n[latex]H_{A}:\\mu_{d}\n<\/tr>\n
      [latex]H_{A}:\\mu_{1}-\\mu_{2}\\neq k[\/latex]<\/td>\n[latex]H_{A}:\\mu_{d}\\neq k[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

      The notations for the summary statistics used to compare paired populations\/samples\u00a0 are shown in the following table. We will use[latex]d[\/latex] to represent the difference variable.<\/p>\n

      \n\n\n\n\n\n\n\n
      Summary Statistics<\/td>\nNotation<\/td>\n<\/tr>\n
      Population Mean of Difference<\/td>\n[latex]\\mu_{d}[\/latex]<\/td>\n<\/tr>\n
      Sample Mean of Difference<\/td>\n[latex]\\bar{d}[\/latex]<\/td>\n<\/tr>\n
      Population Standard Deviation of Difference<\/td>\n[latex]\\sigma_{d}[\/latex]<\/td>\n<\/tr>\n
      Sample Standard Deviation of\u00a0 Difference<\/td>\n[latex]s_{d}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n
      \n

      Question 10<\/h3>\n

      10) It is a common belief that using higher-octane fuel will improve the gas mileage of a vehicle. In order to test this claim, a mechanic randomly selects 12 customers to\u00a0 participate in a study. The mechanic puts 10 gallons of fuel in each participant\u2019s car\u00a0 and asks participants to circle a racetrack until they run out of gas. Each participant\u00a0 is asked to perform this action two times, once with 87-octane fuel and another time\u00a0 with 92-octane fuel. The differences in miles driven (miles driven with 87-octane fuel and miles driven with 92-octane fuel) are calculated and recorded. The\u00a0 participants do not know which fuels they are using while they are driving around\u00a0 the racetrack.<\/p>\n

      What are the appropriate null and alternative hypotheses for this scenario?<\/p>\n<\/div>\n

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