This page titled 9.E: Hypothesis Testing with One Sample (Exercises) is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. Data entry and IT jobs use sample mean values to accomplish daily goals. One Introduction to Statistics class was skeptical. Careers, National Center for Biotechnology Information, Lister Hill National Center for Biomedical Communications, Agency for Healthcare Research and Quality, Centers for Disease Control and Prevention, Robert Wood Johnson Foundation County Health Rankings & Roadmaps, Centers for Medicare and Medicaid Services, http://www.impaqint.com/sites/default/files/research-briefs/Chronic_Conditions_PUF_Brief_Final.pdf. Ninety-five high school and college students I did ask. A survey in the N.Y. Times Almanac finds the mean commute time (one way) is 25.4 minutes for the 15 largest US cities. Type I error: We conclude that the proportion of high school seniors who get drunk each month is not 29%, when it really is 29%. 24. Reason for decision: \(p\text{-value} < \alpha\). After four years the girls were surveyed again. Bethesda, MD 20894, Web Policies (d) What is the mean number of clams per sample? Approximately how many clams would be developing in 10 degree Celsius water? For the example of the phone company, we would find the square root of 2,419,854 to determine the standard error. [Z( %H49WA3P%h@ } To better understand the process of finding a sample mean, consider the steps in terms of the following example. Let \(\bar{X}\) the average I.Q. Type II error: We conclude that the proportion of Americans who prefer to live away from cities is half when, in fact, it is not half. When the test program is run to make an error occur, the likelihood of generating an error is 1%. What is the mean number of clams per sample? They are excellent problems! At the 0.05 significance level, can the survey be considered to be accurate? Type I error: We conclude that the proportion of Americans who prefer to live away from cities is not about half, though the actual proportion is about half. According to an article in Bloomberg Businessweek, New York City's most recent adult smoking rate is 14%. However, youve already done most of the legwork to solve for standard error once youve run these equations. The sample mean is equal to the population mean. At the 5% significance level, there is not enough evidence to conclude that the proportion of homes in Kentucky that are heated by natural gas is 0.517. For each of the word problems, use a solution sheet to do the hypothesis test. Claims data, also known as administrative data, are another sort of electronic record, but on a much bigger scale. _______________________________, 4. 7 0 obj Sixty-three said they smoked to stay thin. HORATIO: In a random survey, for which purpose thou thyself sent me forth, I did discover that one-and-forty believe fervently that the spirits of the dead walk with us. Let \(\bar{X} =\) the mean number of bubbles per blow. The sample mean of the phone companys customer base over six months is 20,792. Find the probability that the sample proportion computed from a sample of size \(900\) will be within \(5\) percentage points of the true population proportion. Confidence Interval: \((0.245, 0.515)\): The plus-4s confidence interval is \((0.259, 0.519)\). To learn what the sampling distribution of \(\hat{p}\) is when the sample size is large. Practice Problem #1 Background: Clams were placed into various temperatures of water. What is the optimum temperature for clam development? While this may not seem much better, I think that I can convince the management to use my test program instead of the original test program. When the sample size is large the sample proportion is normally distributed. A survey of 22 LTCC Intermediate Algebra students generated a mean of 7.24 hours with a standard deviation of 1.93 hours. Private universities' mean tuition cost is more than $20,000 per year. The action you just performed triggered the security solution. HAMLET: That you must teach me. Saying half his friends (including their brothers). Type I error: We conclude that the duration of paid vacations each year for Europeans is not six weeks, when in fact it is six weeks. The organization thinks that, currently, the mean is higher. Claims databases collect information on millions of doctors' appointments, bills, insurance information, and other patient-provider communications. 1. Registered nurses earned an average annual salary of $69,110. A particular brand of tires claims that its deluxe tire averages at least 50,000 miles before it needs to be replaced. The organization thinks that, currently, the mean is higher. To understand the meaning of the formulas for the mean and standard deviation of the sample proportion. Here are the data I wrote in my notebook: I could see that the cost varied but I had to sit down to figure out whether or not I was right. Sum the numbers in your data set and divide the sum by the number of data points that is your mean. (Horatio and Polonius leave together. 1. << /Type /Page /Parent 3 0 R /Resources 6 0 R /Contents 4 0 R /MediaBox [0 0 841.8898 595.2756] Reason for decision: The \(p\text{-value}\) is greater than 0.01. Reason for decision: The \(p\text{-value}\) is greater than 0.05. What does CLAM mean? To understand the meaning of the formulas for the mean and standard deviation of the sample proportion. Legal. 81 women were surveyed with the following results. NOTE: For more accurate results, use more California community colleges and this past year's data. \(H_{0}: \mu \geq 11.52\) \(H_{a}: \mu < 11.52\). You can specify conditions of storing and accessing cookies in your browser. Fifteen randomly chosen teenagers were asked how many hours per week they spend on the phone. State the null and alternative hypotheses. A contingent doubts that the percent is really that high. Conclusion: At the 5% significance level, there is insufficient evidence to conclude that the mean number of sick days is not ten. if in a sample of \(200\) people entering the store, \(78\) make a purchase, \[\hat{p}=\dfrac{78}{200}=0.39. hbbd```b``
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\nonumber\]. In computer science, we use the word clamp as a way to restrict a number between two other numbers. We all get together from different towns. Figure \(\PageIndex{1}\) shows that when \(p = 0.1\), a sample of size \(15\) is too small but a sample of size \(100\) is acceptable. The sample average was $71,121 with a sample standard deviation of $7,489. As a result of the contingents survey, would you agree with the Newsweek poll? which lies wholly within the interval \([0,1]\), so it is safe to assume that \(\hat{p}\) is approximately normally distributed. I had thought that the mean cost of a box (the normal size, not some super-gigantic-family-value-pack) was at most $1, but now I wasn't so sure. From past studies of this tire, the standard deviation is known to be 8,000. Before my God, I might not this believe, without the sensible and true avouch of mine own eyes. (Horatio exits, followed by Polonius, leaving Hamlet to ponder alone. Often sampling is done in order to estimate the proportion of a population that has a specific characteristic, such as the proportion of all items coming off an assembly line that are defective or the proportion of all people entering a retail store who make a purchase before leaving. The class took a random sample of 30 freshman students and found a mean study time of 137 minutes with a standard deviation of 45 minutes. Twelve costs yield a mean cost of 95 with a standard deviation of 18. The formula for solving the sample mean is x#772;=(xi) / n. At first glance of this formula, you may already be ready to throw in the towel, but the complicated seeming language of the equation is actually fairly easy when broken down. { "9.01:_Prelude_to_Hypothesis_Testing" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "9.02:_Null_and_Alternative_Hypotheses" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "9.03:_Outcomes_and_the_Type_I_and_Type_II_Errors" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "9.04:_Distribution_Needed_for_Hypothesis_Testing" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "9.05:_Rare_Events_the_Sample_Decision_and_Conclusion" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", 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At the 5% significance level, there is enough evidence to conclude that the mean amount of summer rain in the northeaster US is less than 11.52 inches, on average. Let \(\bar{X} =\) the mean cost in dollars of macaroni and cheese in a certain town. Conduct a hypothesis test. The more spots, he thought, the more shekels. So my favorite and main staple food is macaroni and cheese. Conclusion: There is sufficient evidence to conclude that the percentage of Americans who have seen or sensed an angel is less than 13%. Sixty-eight percent of online courses taught at community colleges nationwide were taught by full-time faculty. They decide to compare customer base totals for the past six months to get an average of how many people are using their service. Never miss an opportunity thats right for you. At the \(\alpha = 0.05 level\), can it be concluded that the mean rainfall was below the reported average? Use the information in the data. I went to 53 of the closest grocery stores and surveyed the prices of macaroni and cheese. 72 Login . The sample mean was 83; the sample standard deviation was ten. Background: Clams were placed into various temperatures of water. Biology questions and answers. For Americans using library services, the American Library Association claims that at most 67% of patrons borrow books. Let \(\bar{X} =\) the average cost of a daily newspaper. To finish solving for variance, the professor plugs in the new values hes found and solves using the sample mean formula. Following are the names. Analyze end user behavior and Feature adoption Metrics Roll back a failed deployment Analyze regression, Devops: What organizational anti-pattern does DevOps help to address? Conduct a hypothesis test, the Type I error is: It is believed that Lake Tahoe Community College (LTCC) Intermediate Algebra students get less than seven hours of sleep per night, on average. Mathematics of any kind can be initially stressful to many people, but calculating a sample mean is one of the most straightforward calculations youll find in statistics. The student academic groups claim appears to be correct. Type I error: We conclude that the proportion is less than 11%, when it is really at least 11%. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Suppose that a survey is conducted to determine this years rate. The mean work week for engineers in a start-up company is believed to be about 60 hours. Thus the Central Limit Theorem applies to \(\hat{p}\). The mean of a sample distribution will, necessarily, be the same as the mean of the original distribution. We would need to expand our sample base to include these possibilities if we wanted to generalize this claim to the entire nation. Over the past few decades, public health officials have examined the link between weight concerns and teen girls' smoking. This means checking that the interval, \[\left [ \hat{p}-3\sqrt{\frac{\hat{p}(1-\hat{p})}{n}},\, \hat{p}+3\sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \right ]\]. The number 124,750 represents the total number of customers that the phone provider had over the six-month sample size. \(p\text{-value} = 0.000002\) which is almost 0. When calculating the sample mean using the formula, you will plug in the values for each of the symbols. Researchers surveyed a group of 273 randomly selected teen girls living in Massachusetts (between 12 and 15 years old). 5 0 obj While you may vaguely recall sample mean as a distant memory from high school mathematics, its applied across many fields and can be exponentially useful for many reasons. Use the Lap time data for Lap 4 (see [link]) to test the claim that Terri finishes Lap 4, on average, in less than 129 seconds. A random sample of 10 weeks had a mean number of 17.7 infections. Make a line graph of the data. Let \(P =\) the proportion of schoolmates who fear public speaking. HAMLET: I beg to differ; I know of a certainty that five-and-seventy in one hundred of us, condemned to the whips and scorns of time as we are, have gazed upon a spirit of health, or goblin damnd, be their intents wicked or charitable. To begin this process, they must determine the sample mean. Let \(P =\) the proportion of students who feel more enriched as a result of taking Elementary Statistics. Figure \(\PageIndex{2}\) shows that when \(p=0.5\) a sample of size \(15\) is acceptable. A sample of 42 individuals at a shopping mall found that the mean number of visits to a restaurant per week was 2.98 with a standard deviation of 1.68.Find a 90% confidence interval for the mean number of restaurant visits A 90% confidence interval for the mean number of restaurant visits is (Use ascending order. 2 0 obj "Macaroni and Cheese, please!!" Use the Initial Public Offering data (see [link]) to test the claim that the mean offer price was $18 per share. The concept of sample mean will arise early on in learning about statistics. It is but a figment of your imagination! Conclusion: There is sufficient evidence to support the claim that the mean cost of daily papers is $1. What if \(\alpha = 0.01\)? In words, define the random variable for this test. But look tot I charge you, my Lord. Fewer than 5% of adults ride the bus to work in Los Angeles. When youre only looking at a sample group, however, you use the term sample mean. It made me stop and think. If the mean is 20 claims and the standard deviation is 3 claims, what is the probability that in a randomly selected month, the insurance company's claims are less than 26 claims? HAMLET, Prince of Denmark and student of Statistics, HOROTIO, friend to Hamlet and fellow student, Let \(P =\) the proportion of errors generated. CHAT. ), HAMLET: To reject, or not reject, that is the question: whether tis nobler in the mind to suffer the slings and arrows of outrageous statistics, or to take arms against a sea of data, and, by opposing, end them. Does it appear that the mean work week has increased for women at the 5% level? Now, what do you think? 0
The US Department of Energy reported that 51.7% of homes were heated by natural gas. _____ 5. Decision: Do not reject the null hypothesis. Approximately how many clams would be developing in 10 degree Celsius water? To apply this to the example of the phone company, they begin by adding together the number of customers they had each month. The STANDS4 Network . \((0.3829, 0.6171)\). Does the evidence support the claim for Kentucky at the \(\alpha = 0.05\) level in Kentucky? (He turns to Horatio.) Suppose that in a survey of 100 people in a certain town, seven of them suffered from depression or a depressive illness. Course Hero is not sponsored or endorsed by any college or university. Water Temperature (C) Number of Developing Clams 15 72 20 92 25 120 30 140 35 99 40 72 45 36 50 0 1. An online retailer claims that \(90\%\) of all orders are shipped within \(12\) hours of being received. The sample mean was 4.75 hours with a sample standard deviation of 2.0. 4980 0 obj
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A medical investigation claims that the average number of infections per week at a hospital in Westchester is 16.3. Statistics and Probability questions and answers. Conclusion: There is insufficient evidence to conclude that less than 60 percent of her students feel more enriched. it is appropriate to use the normal distribution to compute probabilities related to the sample proportion \(\hat{P}\). In basic terms, the Central Limit Theorem states that as a sample mean becomes larger, it approaches a normal distribution. This page titled 6.3: The Sample Proportion is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. Out of 76 Americans surveyed, only two had seen or sensed the presence of an angel. CLAMS Meaning. Primary \(H_{a}\), in terms of the appropriate parameter \((\mu \text{or} p)\). This means that most of the scores in the class fall within 4.36 points above or below the mean of 85.89. Information and translations of CLAM in the most comprehensive dictionary definitions resource on the web. Finally, the equation asks you to divide by n, which stands for the total number of values that you are comparing. To conclude the drug is safe when, in fact, it is safe. << /ProcSet [ /PDF /ImageB /ImageC /ImageI ] /XObject << /Im1 7 0 R >> >> Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org.
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what is the mean number of clams per sample