Welcome to Westonci.ca, where your questions are met with accurate answers from a community of experts and enthusiasts. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
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.
We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.
