Statistics Unit 5 Milestone Course Hero rated A download to score A

Statistics Unit 5 Milestone Course Hero rated A download to score ASTATISTIC UNIT 5 MILESTONE COURSE HERO 6/29/2020 Sophia :: Welcome If we first note the denominator of Then, getting the z-score we can note it is This tells us that 36.1 is 2.83 standard deviations above the value of 36. Note that when you round some values you may get slightly different results, but the results should be relatively close to this final calculated value. CONCEPT Z-Test for Population Means 2 Emile has calculated a one-tailed z-statistic of -1.97 and wants to see if it is significant at the 5% significance level. What is the critical value for the 5% significance level? Answer choices are rounded to the hundredths place. -2.33 0 -1.04 -1.64 RATIONALE Recall that when a test statistic is smaller than in a left-tailed test we would reject Ho. The closest value to 5%, or 0.05, in the table would be between 0.0505 and 0.495. 0.0505 corresponds with a z-score of -1.64 0.0495 corresponds with a z-score of -1.65. We need to calculate the average of the two z-scores to get a z-score of -1.645. CONCEPT How to Find a Critical Z Value 3 6/29/2020 Sophia :: Welcome What do the symbols , , and represent? Variables of interest Defined variables Population parameters Sample statistics RATIONALE Recall that is the sample proportion, is the sample mean, and is the sample standard deviation. Since all of these come from samples they are statistics. CONCEPT Sample Statistics and Population Parameters 4 A coin is tossed 50 times, and the number of times heads comes up is counted. Which of the following statements about the distributions of counts and proportions is FALSE? The distribution of the count of getting heads can be approximated with a normal distribution. The distribution of the count of getting tails can be approximated with a normal distribution. The count of getting heads is a binomial distribution. The count of getting heads from a sample proportion of size 20 can be approximated with a normal distribution. RATIONALE 6/29/2020 Sophia :: Welcome If we look at the counts from a large population of success and failures (2 outcomes), this is called a binomial distribution, not a normal distribution. CONCEPT Distribution of Sample Proportions 5 Henri has calculated a z-test statistic of -2.73. What is the p-value of the test statistic? Answer choices are rounded to the thousandths place. 0.003 0.004 0.394 6/29/2020 Sophia :: Welcome 0.006 RATIONALE If we go to the chart and the row for the z-column for -2.7 and then the column 0.03, this value corresponds to 0.0032 or 0.003. CONCEPT How to Find a P-Value from a Z-Test Statistic 6 Keith tabulated the values for the average speeds on each day of his road trip as 61.5, 62.2, 55.7, 50.6, 71.3, 70.8, and 66.8 mph. He wishes to construct a 90% confidence interval. What value of t* should Keith use to construct the confidence interval? Answer choices are rounded to the thousandths place. 1.895 6/29/2020 Sophia :: Welcome 1.415 1.943 1.440 RATIONALE Recall that we have n = 7, so the df = n-1 = 6. So if we go to the row where df = 7 and then 0.05 for the tail probability, this gives us a value of 1.943. Recall that a 90% confidence interval would have 10% for the tails, so 5% for each tail. We can also use the last row and find the corresponding confidence level. CONCEPT How to Find a Critical T Value 7 The data below shows the daily low temperatures, in degrees Fahrenheit, of a city for one week. Day Low Temperature, in Fahrenheit Monday 54.5 Tuesday 53 Wednesday 56.5 Thursday 54 Friday 52.5 Saturday 51 Sunday 53 The standard error of the sample mean for this set of data is __________. Answer choices are rounded to the hundredths place. 0.65 0.25 1.31 6/29/2020 Sophia :: Welcome 1.73 RATIONALE In order to get the standard error of the mean, we can use the following formula: , where is the standard deviation and is the sample size. Either calculate by hand or use Excel to find the standard deviation, which is 1.73. The sample size is seven days. The standard error is then: CONCEPT Calculating Standard Error of a Sample Mean 8 A table represents the number of students who passed or failed an aptitude test at two different campuses. South Campus North Campus Passed 42 31 Failed 58 69 In order to determine if there is a significant difference between campuses and pass rate, the chi-square test for association and independence should be performed. What is the expected frequency of South Campus and passed? 36.5 students 42 students 43.7 students 50 students RATIONALE 6/29/2020 Sophia :: Welcome In order to get the expected counts we can note the formula is: CONCEPT Chi-Square Test for Homogeneity 9 A researcher has a table of data with 5 column variables and 4 row variables. The value for the degrees of freedom in order to calculate the statistic is __________. 12 20 19 4 RATIONALE Recall to get the degrees of freedom we use df = (r-1)(c-1) where c and r are the number of rows and columns. This means df = (5-1)(4-1) = 4*3 =12. CONCEPT Chi-Square Test for Association and Independence 10 Edwin conducted a survey to find the percentage of people in an area who smoked regularly. He defined the label “smoking regularly” for males smoking 30 or more cigarettes in a day and for females smoking 20 or more. Out of 635 people who took part in the survey, 71 are labeled as people who smoke regularly. 6/29/2020 Sophia :: Welcome Edwin wishes to construct a significance test for his data. He finds that the proportion of chain smokers nationally is 14.1%. What is the z-statistic for this data? Answer choices are rounded to the hundredths place. -2.11 -0.24 -0.03 -2.34 RATIONALE To make things a little easier, let's first note the denominator We can now note that Finally, subbing all in we find *note that if you round, the values can be slightly different. CONCEPT Z-Test for Population Proportions 11 What value of z* should be used to construct a 97% confidence interval of a population mean? Answer choices are rounded to the thousandths place. 6/29/2020 Sophia :: Welcome 1.65 1.96 1.88 2.17 RATIONALE Using the z-chart to construct a 97% CI, this means that there is 1.5% for each tail. The lower tail would be at 0.015 and the upper tail would be at (1 - 0.015) or 0.985. The value of 0.9850 is actually on the z-table exactly. 0.9850 corresponds with a z-score of 2.17. 6/29/2020 Sophia :: Welcome CONCEPT Confidence Intervals 12 Edwin conducted a survey to find the percentage of people in an area who smoked regularly. He defined the label “smoking regularly” for males smoking 30 or more cigarettes in a day and for females smoking 20 or more. Out of 635 persons who took part in the survey, 71 are labeled as people who smoke regularly. What is the standard error for the sample proportion? Answer choices are rounded to the thousandths place. 2.778 0.112 0.013 2.818 RATIONALE We can note the SE of the proportion is . If we note that , which means . So if we take all this information we can note SE = . CONCEPT Calculating Standard Error of a Sample Proportion 13 Amanda is the owner of a small chain of dental offices. She sent out the yearly satisfaction survey to 600 randomly selected patients and received 544 surveys back. When looking through the results, she noticed that the downtown dental office staff had 84% of clients reporting satisfaction with services, while the uptown dental office staff had 76% of clients reporting satisfaction with services. Which of the following sets shows Amanda's null hypothesis and alternative hypothesis? 6/29/2020 Sophia :: Welcome Null Hypothesis: The proportion of clients satisfied at the downtown office is equal to the proportion of clients satisfied at the uptown office. Alternative Hypothesis: There is a difference in the satisfaction between the uptown and the downtown clients. Null Hypothesis: The proportion of clients satisfied at the downtown office is 84%. Alternative Hypothesis: Uptown clients are more satisfied with the dental office staff than downtown clients. Null Hypothesis: The proportion of clients satisfied at the downtown office is greater than the proportion of clients satisfied at the uptown office. Alternative Hypothesis: Downtown clients are less satisfied with the dental office staff than uptown clients. Null Hypothesis: The proportion of clients satisfied at the uptown office is 76%. Alternative Hypothesis: There is no difference in the satisfaction between the uptown and the downtown clients. RATIONALE Recall that the null hypothesis is always of no difference. So the null hypothesis (Ho) is that the proportion of patients satisfied at the uptown clinic = proportion satisfied at the downtown clinic. This would indicate no difference between the two groups. The alternative hypothesis (Ha) is that there is difference in the proportion of patients satisfied between the two groups. CONCEPT Hypothesis Testing 14 Which of the following assumptions for a two-way ANOVA is FALSE? The samples must be dependent. The sample populations must be normally or approximately normally distributed. 6/29/2020 Sophia :: Welcome The variances of the populations must be equal. The groups must have the same sample size. RATIONALE Inside of the one-way ANOVA we assume that we take an independent and identically distributed sample. So we don't assume dependence. CONCEPT One-Way ANOVA/Two-Way ANOVA 15 The table below shows the results of a customer satisfaction survey at a particular restaurant broken down by males and females. Male Female Extremely Satisfied 25 7 Satisfied 21 13 Neutral 13 16 Dissatisfied 9 14 Extremely Dissatisfied 2 5 Assuming all 5 choices are equally likely, select the observed and expected frequency for male customers that are dissatisfied. Observed: 9 Expected: 25 Observed: 9 Expected: 12.5 Observed: 9 Expected: 14 Observed: 14 6/29/2020 Sophia :: Welcome Expected: 9 RATIONALE If we simply go to the chart then we can directly see the observed frequency for male customers who are dissatisfied is 9. To find the expected frequency, we need to find the number of occurrences if the null hypothesis is true, which in this case, was that the five options are equally likely, or if the five options were all evenly distributed. First, add up all the options in the Male column: If each of these five options were evenly distributed among the 70 males, we would need to divide the total evenly between the five options: This means we would expect 14 men to be extremely satisfied, satisfied, neutral, dissatisfied, and extremely dissatisfied. CONCEPT Chi-Square Statistic 16 Which of the following is an example of a statistic? A local newspaper reporter asks 50 citizens for their opinion on a recent city ordinance. Fifty households randomly sampled in one county have a mean weekly amount spent on groceries. Fifty employees in a small business are randomly selected to take a survey. All 25 students in a third grade class are asked to rank their favorite fruits. RATIONALE A sample is a subset of the entire group of interest. A statistic is something that comes from a sample. Since these 50 households are only a subset of all in the population, this is an example of a statistic from a sample. 6/29/2020 Sophia :: Welcome CONCEPT Sample Statistics and Population Parameters 17 Adam tabulated the values for the average speeds on each day of his road trip as 60.5, 63.2, 54.7, 51.6, 72.3, 70.7, 67.2, and 65.4 mph. The sample standard deviation is 7.309. Select the 98% confidence interval for Adam’s set of data. 46.94 to 71.33 55.45 to 79.46 46.94 to 79.46 55.45 to 70.95 RATIONALE In order to get the 98% CI , we first need to find the critical t-score. Using a t-table, we need to find (n-1) degrees of freedom, or (8-1) = 7 df and the corresponding CI. Using the 98% CI in the bottom row and 7 df on the far left column, we get a t-critical score of 2.998. We also need to calculate the mean: So we use the formula to find the confidence interval: The lower bound is: 63.2-7.75 = 55.45 The upper bound is: 63.2+7.75 = 70.95 CONCEPT Confidence Intervals Using the T-Distribution 6/29/2020 Sophia :: Welcome 18 Edwin conducted a survey to find the percentage of people in an area who smoked regularly. He defined the label “smoking regularly” for males smoking 30 or more cigarettes in a day and for females smoking 20 or more. Out of 635 persons who took part in the survey, 71 are labeled as people who smoke regularly. What is the 90% confidence interval for this population proportion? Answer choices are rounded to the hundredths place. 0.09 to 0.13 0.09 to 0.80 0.11 to 0.80 0.11 to 0.13 RATIONALE In order to get the CI we want to use the following form. First, we must determine the corresponding z*score for 90% Confidence Interval. Remember, this means that we have 5% for the tails, meaning 5%, or 0.05, for each tail. Using a z-table, we can find the upper z-score by finding (1 - 0.05) or 0.95 in the table. This corresponding z-score is at 1.645. We can know So putting it together: 6/29/2020 Sophia :: Welcome The lower bound is: 0.11-0.02 = 0.09 The upper bound is: 0.11 + 0.02 = 0.13 CONCEPT Confidence Interval for Population Proportion 19 One condition for performing a hypothesis test is that the observations are independent. Mary is going to take a sample from a population of 500 students. How many students will Mary have to sample without replacement to treat the observations as independent? RATIONALE In general we want about 10% or less to still assume independence. So size = 0.1*N = 0.1(500) = 50 A sample of 50 or less would be sufficient. CONCEPT 6/29/2020 Sophia :: Welcome Sampling With or Without Replacement 20 The data below shows the grams of fat in a series of popular snacks. Snack Grams of Fat Snack 1 9 Snack 2 13 Snack 3 21 Snack 4 30 Snack 5 31 Snack 6 31 Snack 7 34 Snack 8 25 Snack 9 28 Snack 10 20 If Morris wanted to construct a one-sample t-statistic, what would the value for the degrees of freedom be? 11 9 5 10 RATIONALE The degrees of freedom for a 1 sample t-test are df=n-1 where n is the sample size. In this case, n=10, then df = n-1 = 10-1 = 9. CONCEPT T-Tests 21 Select the statement that correctly describes a Type II error. 6/29/2020 Sophia :: Welcome A Type II error occurs when the null hypothesis is accepted when it is actually false. A Type II error occurs when the null hypothesis is accepted when it is actually true. A Type II error occurs when the null hypothesis is rejected when it is actually false. A Type II error occurs when the null hypothesis is rejected when it is actually true. RATIONALE Recall a Type II error is when we incorrectly accept a false null hypothesis. In this case, we want to reject and conclude there is evidence is correct. CONCEPT Type I/II Errors 22 Rachel recorded the number of calls she made at work during the week: Day Calls Monday 18 Tuesday 14 Wednesday 24 Thursday 16 She expected to make 18 calls each day. To answer whether the number of calls follows a uniform distribution, a chi-square test for goodness of fit should be performed. (alpha = 0.10) What is the chi-squared test statistic? Answers are rounded to the nearest hundredth. 4.61 3.11 6/29/2020 Sophia :: Welcome 3.27 1.52 RATIONALE Using the chi-square formula we can note the test statistic is CONCEPT Chi-Square Test for Goodness-of-Fit 23 Joe hypothesizes that the average age of the population of Florida is less than 37 years. He records a sample mean equal to 37 and states the hypothesis as μ = 37 vs μ < 37. What type of test should Joe do? Left-tailed test Joe should not do any of the types of tests listed Right-tailed test Two-tailed test RATIONALE 6/29/2020 Sophia :: Welcome Since the Ha is a less than sign, this indicates he wants to run a 1 tailed test where the rejection region is the lower or left tail. This can be called a left-tailed test. CONCEPT One-Tailed and Two-Tailed Tests 24 Mrs. Pellegrin has weighed 5 packages of cheese and recorded the weights as 10.2 oz, 10.5 oz, 9.3 oz, 9.8 oz, and 10.0 oz. She calculated the standard deviation to be 0.45 oz. The cheese factory lists the package weight at 10 oz. The t-statistic for a two-sided test would be __________. -0.20 +0.20 -0.40 +0.40 RATIONALE Using the information given, we need to find the sample mean: We now know the following information: Let's plug in the values into the formula: 6/29/2020 Sophia :: Welcome CONCEPT