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Comprehensive Statistical Analysis And Probability Problems

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    Comprehensive Statistical Analysis and Probability Problems Answer all 30 questions. Make sure your answers are as complete as possible. Show all of your work and reasoning. In particular, when there are calculations involved, you must show how you came up with your answers with critical work and/or necessary tables. Answers that come straight from programs or software packages will not be accepted. Refer to the following table for Questions 1, 2, and 3. The table shows temperatures on the first 12 days of October in a small town in Maryland. Date Temperature Date Temperature Date Temperature Oct 1 73 Oct 5 53 Oct 9 62 Oct 2 66 Oct 6 53 Oct 10 49 Oct 3 66 Oct 7 62 Oct 11 52 Oct 4 70 Oct 8 55 Oct 12 57 1. Determine the five number summary for this data. Answer: To determine the five - number summary for the given temperature data, we need to calculate the following: 1. Minimum : The smallest value in the dataset. 2. First Quartile (Q1) : The median of the lower half of the data. 3. Median (Q2) : The middle value when the data is arranged in ascending order. 4. Third Quartile (Q3) : The median of the upper half of the data. 5. Maximum : The largest value in the dataset. Step 1: Organize the data in ascending order. Here are the temperatures from the table: 73, 66, 66, 70, 53, 53, 62, 55, 62, 49, 52, 57. Now, we will sort the data: 49, 52, 53, 53, 55, 57, 62, 62, 66, 66, 70, 73. Step 2: Find the minimum and maximum. • Minimum : 49 • Maximum : 73 Step 3: Find the median (Q2).

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    The median is the middle value of the ordered data. Since there are 12 data points, the median will be the average of the 6th and 7th values. • The 6th value is 57 and the 7th value is 62 . Median (Q2)=57+622=1192=59.5 \ text{Median (Q2)} = \ frac{57 + 62}{2} = \ frac{119}{2} = 59.5 Step 4: Find the first quartile (Q1). The first quartile is the median of the lower half of the data (the first 6 values): 49, 52, 53, 53, 55, 57. The median of these values is the average of the 3rd and 4th values: • The 3rd value is 53 and the 4th value is 53 . Q1=53+532=53 \ text{Q1} = \ frac{53 + 53}{2} = 53 Step 5: Find the third quartile (Q3). The third quartile is the median of the upper half of the data (the last 6 values): 62, 62, 66, 66, 70, 73. The median of these values is the average of the 3rd and 4th values: • The 3rd value is 66 and the 4th value is 66 . Q3=66+662=66 \ text{Q3} = \ frac{66 + 66}{2} = 66 Step 6: Write the five - number summary. Now that we have all the values, the five - number summary is: • Minimum : 49 • Q1 : 53 • Median (Q2) : 59.5 • Q3 : 66 • Maximum : 73 So, the five - number summary is: 49, 53, 59.5, 66, 73 . 2. Determine the mean temperature. Answer: To determine the mean temperature, we need to find the sum of all the temperatures and then divide by the number of days (which is 12 in this case). Step 1: Sum of all the temperatures

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    The temperatures are: 73, 66, 66, 70, 53, 53, 62, 55, 62, 49, 52, 57. Now, let's add them up: 73+66+66+70+53+53+62+55+62+49+52+57=71373 + 66 + 66 + 70 + 53 + 53 + 62 + 55 + 62 + 49 + 52 + 57 = 713 Step 2: Calculate the mean To find the mean, divide the sum by the number of days (12): Mean=71312≈59.42 \ text{Mean} = \ frac{713}{12} \ approx 59.42 Final Answer: The mean temperature is approximately 59.42°F . 3. Determine the mode(s), if any. Answer: The mode is the value that appears most frequently in a dataset. Let's look at the temperatures again: 73, 66, 66, 70, 53, 53, 62, 55, 62, 49, 52, 57. Now, let's count how many times each temperature appears: • 73 appears 1 time. • 66 appears 2 times. • 70 appears 1 time. • 53 appears 2 times. • 62 appears 2 times. • 55 appears 1 time. • 49 appears 1 time. • 52 appears 1 time. • 57 appears 1 time. Step 2: Determine the mode(s) The temperatures that appear the most frequently (2 times each) are: • 66 • 53 • 62 Final Answer:

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    There are three modes in this dataset: 66 , 53 , and 62 . This is called a multimodal distribution. Refer to the following situation for Questions 4, 5, and 6. Consider the following distribution by age groups. Age Frequency Relative Frequency 10 – 14 0.04 15 - 19 20 – 24 10 25 – 29 6 30 – 34 5 35 – 39 4 40 – 44 2 45 – 49 1 Total 50 4. Complete the table. Answer: To complete the table, we need to fill in the missing frequencies and relative frequencies. We are given: • The total frequency is 50. • The relative frequency for the age group 10 - 14 is 0.04. • The frequencies for the age groups 20 - 24, 25 - 29, 30 - 34, 35 - 39, 40 - 44, and 45 - 49 are provided as 10, 6, 5, 4, 2, and 1, respectively. Step 1: Find the missing frequency for the age group 15 - 19. The sum of the frequencies should equal the total frequency, which is 50. So, we calculate the missing frequency for the 15 - 19 age group. We already know the following frequencies: • 20 - 24: 10 • 25 - 29: 6 • 30 - 34: 5 • 35 - 39: 4

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    • 40 - 44: 2 • 45 - 49: 1 Let's add these up: 10+6+5+4+2+1=2810 + 6 + 5 + 4 + 2 + 1 = 28 Now, subtract this from the total frequency (50) to find the frequency for the 15 - 19 age group: 50−28=2250 - 28 = 22 So, the frequency for the 15 - 19 age group is 22. Step 2: Find the relative frequency for each group. Relative frequency is calculated as: Relative frequency=Frequency of the groupTotal frequency \ text{Relative frequency} = \ frac{ \ text{Frequency of the group}}{ \ text{Total frequency}} Complete table: Now the table is complete! Let me know if you'd like more clarification or further help. 5. Is the distribution bell - shaped, skewed to the left, or skewed to the right? Explain your answer. Answer: To determine whether the distribution is bell - shaped, skewed to the left, or skewed to the right, we need to analyze the relative frequencies across the age groups. Here’s a quick rundown of the distribution:

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    • The frequencies are highest in the 15 - 19 age group (22), followed by the 20 - 24 age group (10), and then decrease as the age increases. The frequencies steadily decrease as we move from the younger age groups to the older age groups. Key Observations: • Skewed to the Right : This occurs when the right tail (older ages) is longer or more spread out than the left tail (younger ages), meaning the majority of the data is concentrated in the lower ages, and there are a few higher values. • Bell - shaped : A bell - shaped distribution would show a peak in the middle with frequencies decreasing symmetrically on both sides. • Skewed to the Left : This occurs when the left tail (younger ages) is longer or more spread out than the right tail (older ages), meaning the data is concentrated in the higher ages. In this case: • The distribution has a peak at the 15 - 19 age group and then the frequencies decrease as the age groups increase. • There is a clear drop - off after the 15 - 19 group, especially with very low frequencies in the 40 - 44 (2) and 45 - 49 (1) groups. Conclusion: This distribution is skewed to the right . The higher frequencies are concentrated in the younger age groups (particularly 15 - 19), and the frequencies decrease as the age increases, especially after the 20 - 24 group. This indicates a longer tail on the right side, which is characteristic of a right - skewed distribution. 6. In what age range is the median? Explain your answer. Answer: To determine the median age range, we need to find the age group that contains the middle value of the distribution. The median is the point that divides the distribution into two equal parts, so we must locate the cumulative frequency that corresponds to the 50th percentile (half of the total data, since we have a total of 50 data points). Step 1: Find the cumulative frequencies. We'll calculate the cumulative frequencies for each age group.

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    Step 2: Identify the median position. Since we have a total of 50 data points, the median will be the 25th data point (because 50 ÷ 2 = 25). Step 3: Determine which age group contains the 25th data point. • The cumulative frequency up to the 15 - 19 age group is 24 , so the 25th data point must be in the next age group. • The cumulative frequency up to the 20 - 24 age group is 34 . Since the 25th data point lies between 25 and 34, it falls within the 20 - 24 age group . Conclusion: The median falls within the 20 - 24 age range, because the 25th data point falls within the cumulative frequency of 24 to 34. 7. Your entire gold coin collection consists of 5 coins. Their weights are 2, 3, 8, 7, and 6 ounces respectively. Find the standard deviation of their weights. Show all work. Answer: To find the standard deviation of the gold coin collection, we need to follow these steps: Step 1: Find the mean (average) weight of the coins. The weights of the coins are: 2, 3, 8, 7, and 6 ounces. The formula for the mean is:

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    Mean=Sum of all weightsNumber of coins \ text{Mean} = \ frac{ \ text{Sum of all weights}}{ \ text{Number of coins}} First, sum the weights of the coins: 2+3+8+7+6=262 + 3 + 8 + 7 + 6 = 26 Now, divide by the number of coins (5): Mean=265=5.2 \ text{Mean} = \ frac{26}{5} = 5.2 Step 2: Find the squared deviations from the mean. Next, for each coin, we subtract the mean (5.2) from its weight and then square the result. • For the 2 - ounce coin: (2−5.2)2=(−3.2)2=10.24(2 - 5.2)^2 = ( - 3.2)^2 = 10.24 • For the 3 - ounce coin: (3−5.2)2=(−2.2)2=4.84(3 - 5.2)^2 = ( - 2.2)^2 = 4.84 • For the 8 - ounce coin: (8−5.2)2=(2.8)2=7.84(8 - 5.2)^2 = (2.8)^2 = 7.84 • For the 7 - ounce coin: (7−5.2)2=(1.8)2=3.24(7 - 5.2)^2 = (1.8)^2 = 3.24 • For the 6 - ounce coin: (6−5.2)2=(0.8)2=0.64(6 - 5.2)^2 = (0.8)^2 = 0.64 Step 3: Find the variance. Now, we calculate the variance. The variance is the average of the squared deviations. First, sum all the squared deviations: 10.24+4.84+7.84+3.24+0.64=26.810.24 + 4.84 + 7.84 + 3.24 + 0.64 = 26.8 Now, divide by the number of coins (5) to get the variance: Variance=26.85=5.36 \ text{Variance} = \ frac{26.8}{5} = 5.36 Step 4: Find the standard deviation. The standard deviation is the square root of the variance: Standard Deviation=5.36≈2.32 \ text{Standard Deviation} = \ sqrt{5.36} \ approx 2.32 Final Answer: The standard deviation of the weights of the gold coins is approximately 2.32 ounces . Refer to the following information for Questions 8 and 9.

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    In a recent survey, participants were asked which Apple products influenced them the most. iPod iPhone iPad Total Age under 20 70 100 130 300 Age 20 – 40 160 140 90 390 Age over 40 190 110 50 350 Total 420 350 270 1040 8. What is the probability that the participant was aged 20 – 40 or was influenced by iPad the most? Answer: To calculate the probability that a participant was aged 20 - 40 or was influenced by the iPad the most, we need to use the addition rule of probability . The formula for this rule is: P(A ∪ B)=P(A)+P(B)−P(A∩B)P(A \ cup B) = P(A) + P(B) - P(A \ cap B) Where: • P(A)P(A) is the probability of the participant being aged 20 - 40, • P(B)P(B) is the probability of the participant being influenced by the iPad, • P(A∩B)P(A \ cap B) is the probability of the participant being both aged 20 - 40 and influenced by the iPad. Step 1: Find the probability of being aged 20 - 40 (P(A)P(A)). The total number of participants is 1040. The number of participants aged 20 - 40 is 390. P(A)=Number of participants aged 20 - 40Total participants=3901040P(A) = \ frac{ \ text{Number of participants aged 20 - 40}}{ \ text{Total participants}} = \ frac{390}{1040} P(A)=0.375P(A) = 0.375 Step 2: Find the probability of being influenced by the iPad the most (P(B)P(B)). The number of participants influenced by the iPad the most is the total number of participants who selected iPad, which is 270. P(B)=Number of participants influenced by iPadTotal participants=2701040P(B) = \ frac{ \ text{Number of participants influenced by iPad}}{ \ text{Total participants}} = \ frac{270}{1040} P(B)=0.2596P(B) = 0.2596

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    Step 3: Find the probability of being both aged 20 - 40 and influenced by the iPad the most (P(A∩B)P(A \ cap B)). From the table, the number of participants aged 20 - 40 who were influenced by iPad the most is 90. P(A∩B)=Number of participants aged 20 - 40 and influenced by iPadTotal participants=901040P(A \ cap B) = \ frac{ \ text{Number of participants aged 20 - 40 and influenced by iPad}}{ \ text{Total participants}} = \ frac{90}{1040} P(A∩B)=0.0865P(A \ cap B) = 0.0865 Step 4: Apply the addition rule. Now, use the addition rule to calculate the total probability: P(A ∪ B)=P(A)+P(B)−P(A∩B)P(A \ cup B) = P(A) + P(B) - P(A \ cap B) P(A ∪ B)=0.375+0.2596−0.0865P(A \ cup B) = 0.375 + 0.2596 - 0.0865 P(A ∪ B)=0.5481P(A \ cup B) = 0.5481 Final Answer: The probability that the participant was aged 20 - 40 or was influenced by the iPad the most is approximately 0.5481 or 54.81% . 9. What is the probability that the participant was influenced by iPod the most, given that the participant is “Age Under 20?” Answer: To calculate the probability that the participant was influenced by the iPod the most, given that the participant is aged under 20 , we can use the conditional probability formula : P(iPod ∣ Age under 20)=P(iPod and Age under 20)P(Age under 20)P( \ text{iPod} \ mid \ text{Age under 20}) = \ frac{P( \ text{iPod and Age under 20})}{P( \ text{Age under 20})} Step 1: Find P(Age under 20)P( \ text{Age under 20}). The total number of participants is 1040. The number of participants aged under 20 is 300. P(Age under 20)=3001040=0.2885P( \ text{Age under 20}) = \ frac{300}{1040} = 0.2885 Step 2: Find P(iPod and Age under 20)P( \ text{iPod and Age under 20}). The number of participants aged under 20 who were influenced by the iPod the most is 70. P(iPod and Age under 20)=701040=0.0673P( \ text{iPod and Age under 20}) = \ frac{70}{1040} = 0.0673 Step 3: Calculate the conditional probability.

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