Inference from Sample statistics and margin of error questions

The correct answer is 3,540. According to the table, of 400 voters randomly sampled, the total number of men and women who plan to vote for Candidate A is . The best estimate of the total number of voters in the town who plan to vote for Candidate A is the fraction of voters in the sample who plan to vote for Candidate A,, multiplied by the total voter population of 6000. Therefore, the answer is .

Choice A is correct. It's given that the mean price of a carton of grape tomatoes in Utah was estimated to be \($ 4.23\), with an associated margin of error of \($ 0.08\). It follows that plausible values for this mean price are between \($ 4.23 -$ 0.08\) and \($ 4.23 + $ 0.08\). Therefore, it's plausible that the mean price of a carton of grape tomatoes for all locations that sell this product in Utah is between \($ 4.15\) and \($ 4.31\). Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors. Choice D is incorrect and may result from conceptual or calculation errors.

Choice B is correct. It's given that \(483\) out of \(803\) voters responded that they would vote for Angel Cruz. Therefore, the proportion of voters from the poll who responded they would vote for Angel Cruz is \(483/803\). It’s also given that there are a total of \(6,424\) voters in the election. Therefore, the total number of people who would be expected to vote for Angel Cruz is \(6,424(483/803)\), or \(3,864\). Since \(3,864\) of the \(6,424\) total voters would be expected to vote for Angel Cruz, it follows that \(6,424 -3,864\), or \(2,560\) voters would be expected not to vote for Angel Cruz. The difference in the number of votes for and against Angel Cruz is \(3,864 -2,560\), or \(1,304\) votes. Therefore, if \(6,424\) people vote in the election, Angel Cruz would be expected to win by \(1,304\) votes. Choice A is incorrect. This is the difference in the number of voters from the poll who responded that they would vote for and against Angel Cruz. Choice C is incorrect. This is the total number of people who would be expected to vote for Angel Cruz. Choice D is incorrect. This is the difference between the total number of people who vote in the election and the number of voters from the poll.

Choice C is correct. It’s given that the mean of the data from a random sample of a population is 37, with an associated margin of error of 3. The most appropriate conclusion that can be made is that the mean of the entire population will fall between 37, plus or minus 3. Therefore, the population mean is between and .Choice A is incorrect. While it’s an appropriate conclusion that the population mean is as low as, or 34, it isn’t appropriate to conclude that the population mean is less than 34. Choice B is incorrect. While it’s an appropriate conclusion that the population mean is as high as, or 40, it isn’t appropriate to conclude that the population mean is greater than 40. Choice D is incorrect. It isn’t an appropriate conclusion that the population mean is less than 34 or greater than 40.

Choice D is correct. It’s given that the manager took a random selection of the receipts of 80 customers from a total of 1,500. It’s also given that of those 80 receipts, 20 showed that the customer had purchased fruit. This means that an appropriate estimate of the fraction of customers who purchased fruit is, or . Multiplying this fraction by the total number of customers yields . Therefore, the best estimate for the number of customers who purchased fruit is 375.Choices A and B are incorrect because an exact number of customers can’t be known from taking a random selection. Additionally, choice A may also be the result of a calculation error. Choice C is incorrect and may result from a calculation error.

Choice A is correct. It’s given that the estimate for the proportion of the population that has the characteristic is \(0.49\) with an associated margin of error of \(0.04\). Subtracting the margin of error from the estimate and adding the margin of error to the estimate gives an interval of plausible values for the true proportion of the population that has the characteristic. Therefore, it’s plausible that the proportion of the population that has this characteristic is between \(0.45\) and \(0.53\). Choice B is incorrect. A value less than \(0.45\) is outside the interval of plausible values for the proportion of the population that has the characteristic. Choice C is incorrect. The value \(0.49\) is an estimate for the proportion based on this sample. However, since the margin of error for this estimate is known, the most appropriate conclusion is not that the proportion is exactly one value but instead lies in an interval of plausible values. Choice D is incorrect. A value greater than \(0.53\) is outside the interval of plausible values for the proportion of the population that has the characteristic.

Choice B is correct. It’s given that from the sample of \(20\) employees at the company, \(16\) of the employees are enrolled in exactly three professional development courses this year. Since \((16/20)\) is equal to \(0.80\), or \(80/100\), it follows that \(80 percent sign\) of the employees in the sample are enrolled in exactly three professional development courses this year. Therefore, the best estimate for the percentage of employees at the company who are enrolled in exactly three professional development courses this year is \(80 percent sign\). It’s given that there are a total of \(400\) employees at the company. Therefore, the best estimate of the number of employees at the company who are enrolled in exactly three professional development courses this year is \((80/100)(400)\), or \(320\). Choice A is incorrect. This is the number of employees from the sample who aren't enrolled in exactly three professional development courses this year. Choice C is incorrect. This is the number of employees who weren't selected for the sample. Choice D is incorrect and may result from conceptual or calculation errors.

Choice C is correct. It is given that 34.6% of 26 students in Mr. Camp’s class reported that they had at least two siblings. Since 34.6% of 26 is 8.996, there must have been 9 students in the class who reported having at least two siblings and 17 students who reported that they had fewer than two siblings. It is also given that the average eighth-grade class size in the state is 26 and that Mr. Camp’s class is representative of all eighth-grade classes in the state. This means that in each eighth-grade class in the state there are about 17 students who have fewer than two siblings. Therefore, the best estimate of the number of eighth-grade students in the state who have fewer than two siblings is 17 × (number of eighth-grade classes in the state), or .Choice A is incorrect because 16,200 is the best estimate for the number of eighth-grade students in the state who have at least, not fewer than, two siblings. Choice B is incorrect because 23,400 is half of the estimated total number of eighth-grade students in the state; however, since the students in Mr. Camp’s class are representative of students in the eighth-grade classes in the state and more than half of the students in Mr. Camp’s class have fewer than two siblings, more than half of the students in each eighth-grade class in the state have fewer than two siblings, too. Choice D is incorrect because 46,800 is the estimated total number of eighth-grade students in the state.

Choice A is correct. It's given that based on a random sample from a population, the estimated mean value for a certain variable for the population is \(20.5\), with an associated margin of error of \(1\). This means that it is plausible that the actual mean value of the variable for the population is between \(20.5 -1\) and \(20.5 + 1\). Therefore, the most appropriate conclusion is that it is plausible that the actual mean value of the variable for the population is between \(19.5\) and \(21.5\). Choice B is incorrect. The estimated mean value and associated margin of error describe only plausible values, not all the possible values, for the actual mean value of the variable, so this is not an appropriate conclusion. Choice C is incorrect. The estimated mean value and associated margin of error describe only plausible values for the actual mean value of the variable, not all the possible values of the variable, so this is not an appropriate conclusion. Choice D is incorrect. Since \(20.5\) is the estimated mean value of the variable based on a random sample, the actual mean value of the variable may not be exactly \(20.5\). Therefore, this is not an appropriate conclusion.

Choice C is correct. It’s given that an estimated 38% of sampled students at the school were in support of a menu change, with a margin of error of 5.5%. It follows that the percent of the students at the school who support a menu change is 38% plus or minus 5.5%. The lower bound of this estimation is, or 32.5%. The bound of this estimation is, or 43.5%. Therefore, plausible values of the percent of the students at the school who support a menu change are between 32.5% and 43.5%.Choice A is incorrect. This is the percent of the sampled students at the school who support a menu change. Choices B and D are incorrect and may result from misinterpreting the margin of error.

Choice C is correct. The mean of the 5 samples is trees per acre. The best estimate for the total number of trees in the forest is the product of the mean number of trees per acre in the sample and the total number of acres in the forest. This is (56)(253) = 14,168, which is between 13,500 and 14,500.Choice A is incorrect and may result from multiplying the minimum number of trees per acre in the sample, 45, by the number of acres, 253. Choice B is incorrect and may result from multiplying the median number of trees per acre in the sample, 52, by the number of acres, 253. Choice D is incorrect and may result from multiplying the maximum number of trees per acre in the sample, 73, by the number of acres, 253.

Choice C is correct. It’s given that an estimated \(35 percent sign\) of people in the population support the legislation, with an associated margin of error of \(3 percent sign\). Subtracting and adding the margin of error from the estimate gives an interval of plausible values for the true percentage of people in the population who support the legislation. Therefore, it’s plausible that between \(32 percent sign\) and \(38 percent sign\) of people in this population support the legislation. The corresponding numbers of people represented by these percentages in the population can be calculated by multiplying the total population, \(50,000\), by \(0.32\) and by \(0.38\), which gives \(50,000(0.32)= 16,000\) and \(50,000(0.38)= 19,000\), respectively. It follows that any value in the interval \(16,000\) to \(19,000\) is a plausible value for the total number of people in the population who support the proposed legislation. Of the choices given, only \(16,750\) is in this interval. Choice A is incorrect. This is the number of people in the sample, rather than in the population, who support the legislation. Choice B is incorrect. This is the number of people in the sample who do not support the legislation. Choice D is incorrect. This is a plausible value for the total number of people in the population who do not support the proposed legislation.

Choice D is correct. Sample size is an appropriate reason for the margin of error to change. In general, a smaller sample size increases the margin of error because the sample may be less representative of the whole population.Choice A is incorrect. The margin of error will depend on the size of the sample of recorded votes, not the number of votes that could not be recorded. In any case, the smaller number of votes that could not be recorded for sample A would tend to decrease, not increase, the comparative size of the margin of error. Choice B is incorrect. Since the percent in favor for sample A is the same distance from 50% as the percent in favor for sample B, the percent of favorable responses doesn’t affect the comparative size of the margin of error for the two samples. Choice C is incorrect. If sample A had a larger margin of error than sample B, then sample A would tend to be less representative of the population. Therefore, sample A is not likely to have a larger sample size.

Choice B is correct. It was estimated that 15% of the beads in the bag are red. Since the bag contains 10,000 beads, it follows that there are an estimated red beads. It’s given that the margin of error is 2%, or beads. If the estimate is too high, there could plausibly be red beads. If the estimate is too low, there could plausibly be red beads. Therefore, the most plausible statement of the actual number of red beads in the bag is .Choices A and D are incorrect and may result from misinterpreting the margin of error. It’s unlikely that more than 1,700 beads or fewer than 1,300 beads in the bag are red. Choice C is incorrect because 200 is the margin of error for the number of red beads, not the lower bound of the range of red beads.

Choice B is correct. It’s given that, based on the sample, an estimate of \(12 percent sign\) of all socks produced by the company in a certain week are defective, with an associated margin of error of \(3.62 percent sign\). This estimate, plus or minus the margin of error, gives an interval of plausible values for the actual percent of all socks produced by the company that week that are defective. Subtracting \(3.62 percent sign\) from \(12 percent sign\) yields \(8.38 percent sign\). Adding \(3.62 percent sign\) to \(12 percent sign\) yields \(15.62 percent sign\). Therefore, it is plausible that between \(8.38 percent sign\) and \(15.62 percent sign\) of all socks produced by the company are defective. Choice A is incorrect and may result from conceptual errors. Choice C is incorrect. \(12 percent sign\) is the estimated percent of defective socks based on the sample. However, since the margin of error for this estimate is known, the most appropriate conclusion is not that the percent of defective socks is exactly \(12 percent sign\) but instead that it lies in an interval of plausible percents. Choice D is incorrect and may result from conceptual errors.

Choice B is correct. Let x be the number of people in the entire town that would be expected to name chocolate. Since the sample of 50 people was selected at random, it is reasonable to expect that the proportion of people who named chocolate as their favorite ice-cream flavor would be the same for both the sample and the town population. Symbolically, this can be expressed as . Using cross multiplication, ; solving for x yields 2,083. The choice closest to the value of 2,083 is choice B, 2,100.Choices A, C, and D are incorrect and may be the result of errors when setting up the proportion, solving for the unknown, or incorrectly comparing the choices to the number of people expected to name chocolate, 2,083.

Choice A is correct. It's given that the estimate for the average weight of all boxes of oranges filled by the company in an \(8\)-hour period is \(23.1\) pounds, with an associated margin of error of \(0.19\) pounds. It follows that plausible values for this average weight are between \(23.1 -0.19\) pounds and \(23.1 + 0.19\) pounds. Therefore, plausible values for the average weight of all boxes of oranges filled by the company are between \(22.91\) pounds and \(23.29\) pounds. Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors. Choice D is incorrect and may result from conceptual or calculation errors.

Choice C is correct. A line of best fit is shown in the scatterplot such that as the value of \(x\) increases, the value of \(y\) decreases. It follows that the slope of the line of best fit shown is -. The slope of a line in the xy-plane that passes through the points \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated as \(y_2 -y_1/x_2 -x_1\). The line of best fit shown passes approximately through the points \((0, 8)\) and \((10, 1)\). Substituting \((0, 8)\) for \((x_1, y_1)\) and \((10, 1)\) for \((x_2, y_2)\) in \(y_2 -y_1/x_2 -x_1\) yields the slope of the line being approximately \(1 -8/10 -0\), which is equivalent to \( -7/10\), or \(-0.7\). Therefore, of the given choices, \(-0.7\) is the closest to the slope of this line of best fit. Choice A is incorrect. The line of best fit shown has a -slope, not a positive slope. Choice B is incorrect. The line of best fit shown has a -slope, not a positive slope. Choice D is incorrect and may result from conceptual or calculation errors.

Choice D is correct. The sample of 150 largemouth bass was selected at random from all the largemouth bass in the pond, and since 30% of the fish in the sample weighed more than 2 pounds, it can be concluded that approximately 30% of all largemouth bass in the pond weigh more than 2 pounds.Choices A, B, and C are incorrect. Since the sample contained 150 largemouth bass, of which 30% weighed more than 2 pounds, this result can be generalized only to largemouth bass in the pond, not to all fish in the pond.

Choice C is correct. Because a random sample of the city council was polled, the proportion of the sample who supported the bill is expected to be approximately equal to the proportion of the total city council who supports the bill. Since 6 of the 20 polled, or 30%, supported the bill, it can be estimated that, or 15, city council members support the bill.Choice A is incorrect. This is the number of city council members in the sample who supported the bill. Choice B is incorrect and may result from a computational error. Choice D is incorrect. This is the number of city council members in the sample of city council members who were not polled.

The correct answer is 94. Of those interviewed, “strongly favor” or “somewhat favor” the use of nuclear energy, and interviewees “somewhat oppose” or “strongly oppose” the use of nuclear energy. The difference between the sizes of the two surveyed groups is . The proportion of this difference among the entire group of interviewees is . Because the sample of interviewees was selected at random from US residents, it is reasonable to assume that the proportion of this difference is the same among all US residents as in the sample. Therefore, the best estimate, in millions, of the difference between the number of US residents who somewhat favor or strongly favor the use of nuclear energy and the number of those who somewhat oppose or strongly oppose it is, which to the nearest million is 94.

Choice A is correct. It’s given that \(20 percent sign\) of the students surveyed responded that they intend to enroll in the study program. Therefore, the proportion of students in Spanish club who intend to enroll in the study program, based on the survey, is \(0.20\). Since there are \(55\) total students in Spanish club, the best estimate for the total number of these students who intend to enroll in the study program is \(55(0.20)\), or \(11\). Choice B is incorrect. This is the best estimate for the percentage, rather than the total number, of students in Spanish club who intend to enroll in the study program. Choice C is incorrect. This is the best estimate for the total number of Spanish club students who do not intend to enroll in the study program. Choice D is incorrect. This is the total number of students in Spanish club.

Choice D is correct. The given margin of error of 3% indicates that the actual percent of all US teens who are heavy texters is likely within 3% of the estimate of 30%, or between 27% and 33%. Therefore, it is unlikely, or doubtful, that the percent of all US teens who are heavy texters would be 35%.Choice A is incorrect. The margin of error doesn’t provide any information about the accuracy of reporting in the study. Choice B is incorrect. Based on the estimate and given margin of error, it is unlikely that the percent of all US teens who are heavy texters would be less than 27%, but it is possible. Choice C is incorrect. While the percent of all US teens who are heavy texters is likely between 27% and 33%, any value within this interval is equally likely. We can’t be certain that the value is exactly 33%.