Is there a relation between incidents of child abuse and number of runaway children? A random sample of cities (over 10,000 population) gave the following information about the number of reported incidents of child abuse and the number of runaway children. (Reference: Federal Bureau of Investigation, U.S. Department of Justice.) City 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Abuse cases 38 19 90 45 112 108 61 11 106 50 109 69 111 101 113 Runaways 220 176 558 270 431 656 394 269 307 420 564 708 587 314 774 Use a 1% level of significance to test the claim that there is a monotone-increasing relationship between the ranks of incidents of abuse and number of runaway children. (a) Rank-order abuse using 1 as the largest data value. Also rank-order runaways using 1 as the largest data value. Then construct a table of ranks to be used for a Spearman rank correlation test. Abuse Cases Rank x Runaways Rank y d = x - y d2 City 1 2 3 4 5 6 7 8 10 11 12 13 14 15 Ed² = Compute the sample test statistic. (Use 3 decimal places.)
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
Given Information :
A random sample of cities (over 10,000 population) gave the provided information about the number of reported incidents of child abuse and the number of runways children .
Claim : There is monotone – increasing relationship between the ranks of incidents of abuse and the number of runways children .
The provided data are shown in the table below, along with the corresponding ranks for the variables X and Y .
| Abuse cases (X) | Runways (Y) | Rank(X) | Rank(Y) | d =r (X) - r(Y) | d2 |
| 38 | 220 | 3 | 2 | 1 | 1 |
| 19 | 176 | 2 | 1 | 1 | 1 |
| 90 | 558 | 8 | 10 | -2 | 4 |
| 45 | 270 | 4 | 4 | 0 | 0 |
| 112 | 431 | 14 | 9 | 5 | 25 |
| 108 | 656 | 11 | 13 | -2 | 4 |
| 61 | 394 | 6 | 7 | -1 | 1 |
| 11 | 269 | 1 | 3 | -2 | 4 |
| 106 | 307 | 10 | 5 | 5 | 25 |
| 50 | 420 | 5 | 8 | -3 | 9 |
| 109 | 564 | 12 | 11 | 1 | 1 |
| 69 | 708 | 7 | 14 | -7 | 49 |
| 111 | 587 | 13 | 12 | 1 | 1 |
| 101 | 314 | 9 | 6 | 3 | 9 |
| 113 | 774 | 15 | 15 | 0 | 0 |
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