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Sagot :
Alright, let's solve this step-by-step to determine which of the given probabilities is the greatest for a standard normal distribution, where [tex]\( z \)[/tex] follows a normal distribution with mean 0 and standard deviation 1. The probabilities we need to compare are:
[tex]\[ P(-1.5 \leq z \leq -0.5) \][/tex]
[tex]\[ P(-0.5 \leq z \leq 0.5) \][/tex]
[tex]\[ P(0.5 \leq z \leq 1.5) \][/tex]
[tex]\[ P(1.5 \leq z \leq 2.5) \][/tex]
To find these probabilities, we need to calculate the cumulative distribution function (CDF) values at different z-scores and use these to find the areas under the normal curve between the specified limits.
### Step 1: Calculate Individual Probabilities
1. For [tex]\( P(-1.5 \leq z \leq -0.5) \)[/tex]:
[tex]\[ P(-1.5 \leq z \leq -0.5) = \Phi(-0.5) - \Phi(-1.5) \][/tex]
2. For [tex]\( P(-0.5 \leq z \leq 0.5) \)[/tex]:
[tex]\[ P(-0.5 \leq z \leq 0.5) = \Phi(0.5) - \Phi(-0.5) \][/tex]
3. For [tex]\( P(0.5 \leq z \leq 1.5) \)[/tex]:
[tex]\[ P(0.5 \leq z \leq 1.5) = \Phi(1.5) - \Phi(0.5) \][/tex]
4. For [tex]\( P(1.5 \leq z \leq 2.5) \)[/tex]:
[tex]\[ P(1.5 \leq z \leq 2.5) = \Phi(2.5) - \Phi(1.5) \][/tex]
Here, [tex]\(\Phi(z)\)[/tex] denotes the cumulative distribution function for the standard normal distribution evaluated at [tex]\(z\)[/tex].
### Step 2: List the Calculated Probabilities
Based on the calculations performed:
- [tex]\( P(-1.5 \leq z \leq -0.5) = 0.2417303374571288 \)[/tex]
- [tex]\( P(-0.5 \leq z \leq 0.5) = 0.38292492254802624 \)[/tex]
- [tex]\( P(0.5 \leq z \leq 1.5) = 0.2417303374571288 \)[/tex]
- [tex]\( P(1.5 \leq z \leq 2.5) = 0.060597535943081926 \)[/tex]
### Step 3: Compare the Probabilities
Now, we compare the obtained probabilities:
- [tex]\( 0.2417303374571288 \)[/tex] (for [tex]\( P(-1.5 \leq z \leq -0.5) \)[/tex])
- [tex]\( 0.38292492254802624 \)[/tex] (for [tex]\( P(-0.5 \leq z \leq 0.5) \)[/tex])
- [tex]\( 0.2417303374571288 \)[/tex] (for [tex]\( P(0.5 \leq z \leq 1.5) \)[/tex])
- [tex]\( 0.060597535943081926 \)[/tex] (for [tex]\( P(1.5 \leq z \leq 2.5) \)[/tex])
Among these probabilities, the greatest probability is:
[tex]\[ P(-0.5 \leq z \leq 0.5) = 0.38292492254802624 \][/tex]
### Conclusion
Hence, the interval [tex]\( P(-0.5 \leq z \leq 0.5) \)[/tex] has the greatest probability for a standard normal distribution.
[tex]\[ P(-1.5 \leq z \leq -0.5) \][/tex]
[tex]\[ P(-0.5 \leq z \leq 0.5) \][/tex]
[tex]\[ P(0.5 \leq z \leq 1.5) \][/tex]
[tex]\[ P(1.5 \leq z \leq 2.5) \][/tex]
To find these probabilities, we need to calculate the cumulative distribution function (CDF) values at different z-scores and use these to find the areas under the normal curve between the specified limits.
### Step 1: Calculate Individual Probabilities
1. For [tex]\( P(-1.5 \leq z \leq -0.5) \)[/tex]:
[tex]\[ P(-1.5 \leq z \leq -0.5) = \Phi(-0.5) - \Phi(-1.5) \][/tex]
2. For [tex]\( P(-0.5 \leq z \leq 0.5) \)[/tex]:
[tex]\[ P(-0.5 \leq z \leq 0.5) = \Phi(0.5) - \Phi(-0.5) \][/tex]
3. For [tex]\( P(0.5 \leq z \leq 1.5) \)[/tex]:
[tex]\[ P(0.5 \leq z \leq 1.5) = \Phi(1.5) - \Phi(0.5) \][/tex]
4. For [tex]\( P(1.5 \leq z \leq 2.5) \)[/tex]:
[tex]\[ P(1.5 \leq z \leq 2.5) = \Phi(2.5) - \Phi(1.5) \][/tex]
Here, [tex]\(\Phi(z)\)[/tex] denotes the cumulative distribution function for the standard normal distribution evaluated at [tex]\(z\)[/tex].
### Step 2: List the Calculated Probabilities
Based on the calculations performed:
- [tex]\( P(-1.5 \leq z \leq -0.5) = 0.2417303374571288 \)[/tex]
- [tex]\( P(-0.5 \leq z \leq 0.5) = 0.38292492254802624 \)[/tex]
- [tex]\( P(0.5 \leq z \leq 1.5) = 0.2417303374571288 \)[/tex]
- [tex]\( P(1.5 \leq z \leq 2.5) = 0.060597535943081926 \)[/tex]
### Step 3: Compare the Probabilities
Now, we compare the obtained probabilities:
- [tex]\( 0.2417303374571288 \)[/tex] (for [tex]\( P(-1.5 \leq z \leq -0.5) \)[/tex])
- [tex]\( 0.38292492254802624 \)[/tex] (for [tex]\( P(-0.5 \leq z \leq 0.5) \)[/tex])
- [tex]\( 0.2417303374571288 \)[/tex] (for [tex]\( P(0.5 \leq z \leq 1.5) \)[/tex])
- [tex]\( 0.060597535943081926 \)[/tex] (for [tex]\( P(1.5 \leq z \leq 2.5) \)[/tex])
Among these probabilities, the greatest probability is:
[tex]\[ P(-0.5 \leq z \leq 0.5) = 0.38292492254802624 \][/tex]
### Conclusion
Hence, the interval [tex]\( P(-0.5 \leq z \leq 0.5) \)[/tex] has the greatest probability for a standard normal distribution.
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