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To find out whether 0 murders in a single day is a significantly low number of murders in a city with 635 murders in a year, follow these steps:
### Step 1: Calculate the Mean Number of Murders per Day
First, we need to determine the mean number of murders per day. Given that there were 635 murders in a year, and a year has 365 days, we can find the mean as follows:
- Total number of murders in a year: 635
- Number of days in a year: 365
Now, compute the mean number of murders per day by dividing the total murders by the number of days in a year:
[tex]\[ \text{Mean murders per day} = \frac{\text{Total murders}}{\text{Number of days}} = \frac{635}{365} \approx 1.7 \][/tex]
So, the mean number of murders per day is approximately 1.7 (rounded to one decimal place).
### Step 2: Calculate the Probability of No Murders in a Single Day
Next, we want to find the probability of having 0 murders in a single day when the mean number of murders per day is 1.7. This situation typically follows a Poisson distribution, often used for the number of events happening in a fixed interval of time.
The probability [tex]\( P(X = k) \)[/tex] of having [tex]\( k \)[/tex] events (murders, in this case) given a mean rate of [tex]\( \lambda \)[/tex] events per interval is given by the Poisson formula:
[tex]\[ P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} \][/tex]
Where:
- [tex]\( \lambda \)[/tex] is the mean rate (1.7 in our case).
- [tex]\( k \)[/tex] is the number of occurrences (0 for no murders).
- [tex]\( e \)[/tex] is the base of the natural logarithm, approximately equal to 2.71828.
For [tex]\( k = 0 \)[/tex]:
[tex]\[ P(X = 0) = \frac{1.7^0 e^{-1.7}}{0!} = e^{-1.7} \][/tex]
Using [tex]\( e^{-1.7} \approx 0.17557 \)[/tex], the probability of 0 murders in a single day is approximately 0.1756 (rounded to four decimal places).
### Step 3: Determine if 0 Murders is Significantly Low
To determine if 0 murders in a single day is significantly low, we typically compare the probability to a threshold such as 0.05 (5%). If the probability is less than 0.05, it is considered significantly low.
In this case, the probability of having 0 murders in a single day is approximately 0.1756, which is greater than 0.05. Therefore, 0 murders in a single day is not considered significantly low.
### Summary
- The mean number of murders per day is approximately 1.7.
- The probability of having no murders in a single day is approximately 0.1756.
- 0 murders in a single day would not be considered a significantly low number.
### Step 1: Calculate the Mean Number of Murders per Day
First, we need to determine the mean number of murders per day. Given that there were 635 murders in a year, and a year has 365 days, we can find the mean as follows:
- Total number of murders in a year: 635
- Number of days in a year: 365
Now, compute the mean number of murders per day by dividing the total murders by the number of days in a year:
[tex]\[ \text{Mean murders per day} = \frac{\text{Total murders}}{\text{Number of days}} = \frac{635}{365} \approx 1.7 \][/tex]
So, the mean number of murders per day is approximately 1.7 (rounded to one decimal place).
### Step 2: Calculate the Probability of No Murders in a Single Day
Next, we want to find the probability of having 0 murders in a single day when the mean number of murders per day is 1.7. This situation typically follows a Poisson distribution, often used for the number of events happening in a fixed interval of time.
The probability [tex]\( P(X = k) \)[/tex] of having [tex]\( k \)[/tex] events (murders, in this case) given a mean rate of [tex]\( \lambda \)[/tex] events per interval is given by the Poisson formula:
[tex]\[ P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} \][/tex]
Where:
- [tex]\( \lambda \)[/tex] is the mean rate (1.7 in our case).
- [tex]\( k \)[/tex] is the number of occurrences (0 for no murders).
- [tex]\( e \)[/tex] is the base of the natural logarithm, approximately equal to 2.71828.
For [tex]\( k = 0 \)[/tex]:
[tex]\[ P(X = 0) = \frac{1.7^0 e^{-1.7}}{0!} = e^{-1.7} \][/tex]
Using [tex]\( e^{-1.7} \approx 0.17557 \)[/tex], the probability of 0 murders in a single day is approximately 0.1756 (rounded to four decimal places).
### Step 3: Determine if 0 Murders is Significantly Low
To determine if 0 murders in a single day is significantly low, we typically compare the probability to a threshold such as 0.05 (5%). If the probability is less than 0.05, it is considered significantly low.
In this case, the probability of having 0 murders in a single day is approximately 0.1756, which is greater than 0.05. Therefore, 0 murders in a single day is not considered significantly low.
### Summary
- The mean number of murders per day is approximately 1.7.
- The probability of having no murders in a single day is approximately 0.1756.
- 0 murders in a single day would not be considered a significantly low number.
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