The Following Table Lists The Number Of Deaths By Cause

The Following Table Lists The Number Of Deaths By Cause As Reported

The assignment involves analyzing data related to causes of death, survey results, median household incomes across states, characteristics of the normal distribution, z-scores, and probabilities related to the GMAT scores. Your task includes identifying variables, constructing frequency distributions, calculating relative frequencies and percentages, drawing bar graphs, constructing class intervals, analyzing distribution shape, interpreting areas under the normal distribution curve, and calculating probabilities using z-scores.

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The data presented in the first part of the assignment pertains to the number of deaths caused by various health conditions, as reported by the Centers for Disease Control and Prevention in 2015. The key statistical concepts involved include understanding the variables, observations, and elements within the data set. The variable for this data set is primarily the cause of death, which categorizes each death by its attributed disease or condition. Specifically, the question asks whether the variable is "the cause of death from each disease" or "the number of deaths from each disease," etc. The appropriate answer is option C: "The cause of death," as it signifies the categorical variable, while the number of deaths is a numerical value associated with each cause.

The data set contains a total of eight observations, corresponding to the eight categories listed: Heart disease, Cancer, Accidents, Stroke, Alzheimer's disease, Diabetes, Influenza and Pneumonia, and Suicide. Each of these categories represents a single data point or observation. The total number of elements in this data set is also eight, as each category has associated death counts. This understanding is crucial for statistical analysis, as the number of observations defines the sample size, and the elements refer to individual data points within that sample.

The second part of the assignment involves analyzing survey data with categorical variables represented by letters A, B, and C. The goal is to create a frequency distribution table, compute relative frequencies and percentages, interpret the distribution shape, and visualize the data graphically.

The frequency distribution table for categories A, B, and C reveals how often each category appears in the sample. Suppose, for example, category A appears 7 times, B appears 9 times, and C appears 8 times. The relative frequency for each category is calculated by dividing the frequency of that category by the total number of observations. For instance, if total observations are 24, then the relative frequency for category A is 7/24 ≈ 0.292, for B is 9/24 ≈ 0.375, and for C is 8/24 ≈ 0.333. Calculating the percentages involves multiplying these relative frequencies by 100 and rounding to one decimal place, giving approximate percentages of 29.2% for A, 37.5% for B, and 33.3% for C.

To determine what percentage of elements belong to Category B, we note the frequency of B and divide by the total. Using the example numbers, approximately 37.5% of the elements are in Category B. Graphically, a bar graph can be created with categories A, B, and C along the x-axis and their corresponding frequencies along the y-axis, illustrating the distribution visually.

The third part relates to median household income data across the 50 states and the District of Columbia. Constructing a frequency distribution table involves selecting class intervals such as $37,000–$41,999, $42,000–$46,999, etc., and tallying how many states fall within each interval based on their median income. For example, states with median incomes from $42,000 to $46,999 may number around 16. The resulting table provides a clear overview of income distribution across states.

Subsequently, calculating relative frequencies entails dividing the frequency of each class by the total number of states (50), and then converting these to percentages. For example, if 16 states fall within the $42,000–$46,999 range, the relative frequency would be 16/50 = 0.32, and the percentage would be 32%. Rounding to one decimal point yields 32.0%. This analysis can help determine whether the income data are symmetrically distributed or skewed. Based on the frequencies, the data may appear right-skewed if higher income classes contain fewer states, or symmetrical if the distribution centers around a middle class.

Finally, calculating the percentage of states with median incomes of $42,000 or more involves summing the frequencies of all classes with lower bounds of $42,000 and higher, then dividing by 50. For example, summing classes $42,000–$46,999, $47,000–$51,999, etc., and dividing by 50 provides the proportion of states in this income range, which then can be converted to a percentage.

The next question addresses characteristics of the normal distribution. It is incorrect to say that the value of the mean is always greater than the standard deviation; in fact, the mean can be less than, equal to, or greater than the standard deviation, depending on the distribution. The other options correctly describe properties such as tails extending indefinitely, symmetry, and the total area under the curve being one.

Regarding the normal distribution, the area to the left of the mean (which is also the median) is always 0.5, signifying 50% of the data lies below the mean. For the standard normal distribution, the mean is zero and the standard deviation is one, a fundamental property that simplifies probability calculations. Using z-scores, the area between z = 0 and z = 2.38 corresponds to the cumulative probability from the mean to that z-value, which can be obtained from standard z-tables or statistical software, typically around 0.9913.

Similarly, the area to the right of z = -1.21 can be found by recognizing that the cumulative area to the left of that z-score is approximately 0.1131, hence the area to the right is 1 - 0.1131 ≈ 0.8869. For the interval between z = 0.19 and z = 0.92, the areas can be derived by subtracting cumulative probabilities; for example, 0.4236 for z = 0.92 and 0.4620 for z = 0.19, making the difference approximately 0.0384.

The last question involves calculating probabilities related to GMAT scores, assuming a normal distribution with a mean of 410 and a standard deviation of 49. To find the probability of scores between 400 and 480, convert these raw scores to z-scores: z = (X – μ)/σ. For 400: z = (400 – 410)/49 ≈ -0.204; for 480: z = (480 – 410)/49 ≈ 1.429. Using standard normal tables, the cumulative probabilities at these z-scores are approximately 0.419 and 0.923, respectively. The probability of a score between 400 and 480 is the difference: 0.923 – 0.419 ≈ 0.5044. Similarly, for scores less than 370, z = (370 – 410)/49 ≈ -0.816, with a cumulative probability around 0.207. For scores greater than 530, z = (530 – 410)/49 ≈ 2.45, with a cumulative probability about 0.9929; thus, the probability of scores above 530 is 1 – 0.9929 ≈ 0.0071.

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