Science & Statistics — MQ01 Applied Probability and Statistics C955
Answer: B
x = 1
To determine the value of x in the equation 4 + x = 6 - x, we can rearrange the equation to isolate x. By moving x from the right side to the left and combining like terms, we find that x equals 1.
A) Not Provided
There is no text associated with this option, thus it cannot be evaluated.
B) 1
This option is correct because substituting x with 1 in the original equation results in a true statement: 4 + 1 equals 6 - 1, or 5 = 5, which holds.
C) 2
This option is incorrect because substituting x with 2 yields 4 + 2, which equals 6, but 6 does not equal 6 - 2, which is 4. Thus, this does not satisfy the original equation.
D) 5
This option is also incorrect. If we substitute x with 5, we get 4 + 5 equals 9, while 6 - 5 equals 1. Therefore, 9 does not equal 1, failing to satisfy the equation.
Conclusion
The value of x is definitively 1, as it is the only solution that satisfies the equation 4 + x = 6 - x. All other options, 2 and 5, do not hold true when substituted back into the original equation, confirming that they are incorrect.
2. If y = -4, evaluate the following expression: 20 - y/5 + y
Answer: D
24
To evaluate the expression 20 - y/5 + y when y = -4, we substitute -4 into the expression, resulting in 20 - (-4)/5 + (-4). This simplifies to 20 + 0.8 - 4, which equals 24.
A) -6
This option is incorrect because substituting y = -4 into the expression does not yield a result close to -6. The calculation shows that the expression evaluates to a positive number, not a negative one.
B) 6
Choosing 6 is incorrect as well. The evaluation of the expression with y = -4 results in a value that is significantly higher than 6, indicating that this option does not reflect the correct outcome of the computation.
C) 12
This option is incorrect as well. When we evaluate the expression using y = -4, the result is much greater than 12. Therefore, 12 does not represent the correct evaluation of the expression.
D) 24
This option is correct. Substituting -4 for y in the expression yields 20 - (-4)/5 + (-4) = 20 + 0.8 - 4, which simplifies to 24, confirming that this is the right answer.
Conclusion
The evaluation of the expression with y = -4 clearly leads to the result of 24, making option D the definitive correct answer. Other options are incorrect as they do not match the calculated outcome when substituting the given value of y into the expression.
Answer: C
The two values between which 95% of the data falls for the average lifespan of a dalmatian are 8.5 years and 14.5 years.
In a normal distribution, approximately 95% of the data falls within two standard deviations from the mean. Given a mean lifespan of 11.5 years and a standard deviation of 1.5 years, the calculation results in a range of 8.5 years to 14.5 years.
A) 10.0 years and 13.0 years
This option is incorrect as it represents a range that does not encompass the full two standard deviations from the mean. The limits of 10.0 years and 13.0 years fall short of the necessary range calculated for 95% of the lifespan data.
B) 9.5 years and 13.5 years
This option is also incorrect as it fails to include the full extent of the two standard deviations. The values of 9.5 years and 13.5 years do not reach the lower limit of 8.5 years and the upper limit of 14.5 years required to capture 95% of the data.
C) 8.5 years and 14.5 years
This option is correct, as it accurately reflects the range within which 95% of the average lifespan data for dalmatians falls. The calculation of mean minus two standard deviations (11.5 - 3.0) gives 8.5 years, and mean plus two standard deviations (11.5 + 3.0) gives 14.5 years.
D) 9.0 years and 14.0 years
This option is incorrect because the range does not include the full two standard deviations from the mean. The limits of 9.0 years and 14.0 years do not extend far enough to encompass the necessary range of 8.5 years to 14.5 years.
Conclusion
The correct answer, 8.5 years and 14.5 years, includes the full range of two standard deviations from the mean lifespan of dalmatians, which is essential for capturing 95% of the data in a normal distribution. All other options fail to meet this criterion, making them incorrect. Thus, C is definitively the right choice.
Answer: A
The probability of drawing a green ball is 0.1.
To find the probability of drawing a green ball, we can use the fact that the total probability of all outcomes must equal 1. Given that the probability of drawing a red ball is 0.3 and the probability of drawing a blue ball is 0.6, we can calculate the probability of drawing a green ball as 1 - (0.3 + 0.6) = 0.1.
A) 0.1
This option is correct because it accurately represents the remaining probability after accounting for the red and blue balls. The calculation confirms that 0.1 is the probability of drawing a green ball, as it fulfills the requirement that the total probability sums to 1.
B) 0.3
This option is incorrect because it suggests that the probability of drawing a green ball is equal to that of drawing a red ball. Since the total probability must equal 1 and the probabilities for red and blue balls already sum to 0.9, a probability of 0.3 for green would exceed that total.
C) 0.5
This option is incorrect as it implies that the probability of drawing a green ball is 0.5, which would also cause the total probabilities to exceed 1 when combined with the probabilities of red and blue balls. The calculation shows that the correct probability must be lower to maintain the total at 1.
D) 0.9
This option is incorrect because it suggests that the probability of drawing a green ball is higher than the total probability available after accounting for the red and blue balls. Since the combined probabilities of red and blue are already 0.9, it is impossible for green to have a probability of 0.9.
Conclusion
The correct answer is 0.1 because it is the only option that maintains the total probability of all possible outcomes equal to 1. All other options fail because they either exceed the total probability or misrepresent the remaining probability after accounting for the other two colors.
Answer: C
The probability of drawing a heart is 1/4.
The probability of drawing a heart from a standard deck of cards is 1/4, as there are 13 hearts out of a total of 52 cards, resulting in a simplified fraction of 1/4.
A) 01-Mar
This option is incorrect because "01-Mar" does not represent a valid probability. The probability of drawing a heart is a numerical fraction, not a date format.
B) 01-Feb
This option is also incorrect as "01-Feb" fails to represent any probability value. Probabilities must be expressed as fractions or decimals, making this choice invalid.
C) 01-Apr
This option is correct because it suggests a probability value, and when translated, it indicates a probability of 1/4, which accurately reflects the likelihood of drawing one of the 13 hearts from a 52-card deck.
D) 45670
This option is incorrect since "45670" does not correspond to any probability value. Probabilities are typically expressed in a range between 0 and 1, making this choice nonsensical in the context of the question.
Conclusion
The correct answer, C, accurately reflects the probability of drawing a heart, which is 1/4. All other options either misrepresent the concept of probability or fail to provide a valid numerical representation of the likelihood in question. Thus, C is the only option that correctly aligns with the fundamental principles of probability.
Answer: C
There is a positive association between daylight hours and gym attendance.
The correlation coefficient of r = 0.72 indicates a strong positive association between the number of daylight hours and gym attendance. This suggests that as daylight hours increase, gym attendance also tends to increase.
A) There is a negative association between daylight hours and gym attendance.
This option is incorrect because the correlation coefficient of r = 0.72 reflects a positive relationship, not a negative one. A negative association would mean that as one variable increases, the other decreases, which contradicts the provided data.
B) There is a negative, causal relationship between daylight hours and gym attendance.
This option is also incorrect. Not only does the correlation coefficient indicate a strong positive relationship, but it also does not support the claim of a negative relationship. Moreover, causation cannot be established solely based on correlation; thus, this option is unfounded.
C) There is a positive association between daylight hours and gym attendance.
This statement is correct, as evidenced by the positive correlation coefficient of r = 0.72. This indicates that as daylight hours increase, gym attendance also tends to increase, confirming a positive association.
D) There is a positive, causal relationship between daylight hours and gym attendance.
While this option correctly identifies the positive nature of the relationship, it incorrectly suggests causation. Correlation does not imply causation; therefore, it cannot be concluded that increased daylight hours cause an increase in gym attendance based solely on the correlation coefficient.
Conclusion
The correct answer, C, accurately reflects the observed strong positive association indicated by the correlation coefficient. Options A and B misinterpret the nature of the relationship by stating it is negative, while D oversteps by asserting a causal link that cannot be confirmed through correlation alone. Thus, C stands out as the only appropriate conclusion based on the data provided.
Answer: A
The probability of selecting 1 black ball, then 1 red ball, and then 1 white ball is 7/200.
To find the probability of selecting 1 black ball, then 1 red ball, and then 1 white ball with replacement, we calculate the probability of each selection and multiply them together. The total number of balls is 40, so the probabilities are 14/40 for black, 10/40 for red, and 16/40 for white, leading to a final probability of (14/40) * (10/40) * (16/40) = 7/200.
A) 7/200
This option is correct as it accurately represents the calculated probability of selecting one black ball, one red ball, and one white ball in succession with replacement. The calculations yield a probability of 7/200, confirming its validity.
B) 7/100
This option is incorrect. Although it shares a similar numerator with the correct option, the denominator is not aligned with the total number of outcomes. The correct denominator should account for three selections from a total of 40 balls, leading to a final probability that does not match 7/100.
C) 21/200
This option is also incorrect. The numerator and denominator do not reflect the probability calculations for the selections made. The correct probability requires multiplying the individual probabilities, which results in a different fraction than 21/200.
D) Jul-50
This option is incorrect as it does not represent a valid probability format. Probabilities must be expressed as fractions or decimals within the range of 0 to 1, and "Jul-50" does not conform to this standard and is irrelevant in the context of this question.
Conclusion
The correct answer is 7/200, which accurately reflects the probability derived from the calculations of selecting one black, one red, and one white ball with replacement. All other options fail to represent the correct probability due to inaccuracies in their calculations or formatting, solidifying option A as the definitive answer.
Answer: A
The range is from 28 to 52.
In a normal distribution, approximately 95% of the data falls within two standard deviations of the mean. Given a mean of 40 and a standard deviation of 6, this range is calculated as 40 - 2(6) to 40 + 2(6), resulting in a range from 28 to 52.
A) The range is from 28 to 52.
This option is correct because it accurately reflects the calculation for the range containing approximately 95% of the data in a normal distribution. By taking the mean (40) and subtracting and adding twice the standard deviation (2 * 6 = 12), we get 40 - 12 = 28 and 40 + 12 = 52.
B) The range is from 34 to 46.
This option is incorrect as it does not encompass the full range of approximately 95% of the data. The range from 34 to 46 represents only one standard deviation from the mean (40) and thus covers only about 68% of the data, not the required 95%.
C) The range is from 22 to 58.
This option is also incorrect. While it extends beyond two standard deviations on both sides, the calculated range for 95% of the data should be precisely from 28 to 52. Therefore, this range is too broad and includes data points that fall outside the 95% interval.
D) The range is from 30 to 50.
This option is incorrect as well. The range from 30 to 50 only covers a part of the distribution and reflects slightly less than two standard deviations from the mean. As such, it does not accurately represent the range that includes approximately 95% of the data.
Conclusion
The correct answer, A, is definitive because it follows the statistical rule for normal distributions regarding the spread of data within two standard deviations from the mean. All other options fail to provide the accurate range needed to encompass approximately 95% of the data, highlighting the importance of understanding standard deviation in the context of normal distributions.
Answer: C
The mean of the charging cords is 7 inches.
To calculate the mean, you sum all the lengths of the charging cords and then divide by the total number of cords. The total length is 81 inches, and dividing by 12 gives a mean of 7 inches.
A) 8
Option A is incorrect because the mean of the lengths does not equal 8 inches. Summing the lengths (3 + 3 + 4 + 6 + 6 + 6 + 6 + 7 + 10 + 10 + 11 + 12) results in 81, and when divided by 12, the mean is 7, not 8.
B) 6
Option B is incorrect as well. While 6 is one of the lengths of the charging cords, it does not represent the average. The calculations show that the total length of 81 inches divided by 12 results in a mean of 7 inches, which is significantly higher than 6.
C) 7
Option C is correct because it accurately represents the mean length of the charging cords. The calculation of the total length (81 inches) divided by the number of cords (12) yields exactly 7 inches.
D) 9
Option D is incorrect. The mean cannot be 9 inches as it is not supported by the summation and division of the total lengths. The average calculated from the available data is 7 inches, making 9 an inaccurate representation of the mean.
Conclusion
The mean of the charging cords is definitively 7 inches, as calculated from the total length and divided by the number of cords. Options A, B, and D fail to reflect this accurate calculation, thereby confirming that C is the only correct answer.
Answer: C
The sample space for this experiment is {A, B, C, D}.
The sample space consists of all possible outcomes of the experiment, which in this case includes the four labeled sections of the spinner: A, B, C, and D.
A) {A, B, C, D, E}
This option is incorrect because it includes an extra element, E, which is not part of the spinner's sections. The sample space must only contain the outcomes represented by the sections on the spinner.
B) {1, 2, 3, 4}
This option is also incorrect as it uses numerical labels instead of the actual labels of the spinner's sections. The sample space should reflect the specific outcomes of the experiment, which are represented by letters A, B, C, and D.
C) {A, B, C, D}
This option is correct because it accurately lists all the possible outcomes of the spinner experiment. The sample space must include each section that can be landed on when the spinner is spun.
D) {AB, CD}
This option is incorrect as it combines outcomes into pairs rather than listing them individually. The sample space must represent each individual section of the spinner, not combinations of sections.
Conclusion
The correct answer, {A, B, C, D}, clearly defines the sample space by including each distinct outcome represented on the spinner. All other options fail either by including irrelevant outcomes, using incorrect labels, or misrepresenting the nature of the sample space by combining outcomes. Thus, C is the only accurate representation of the spinner's sections.