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Sagot :
To determine which statement is equivalent to [tex]\( P(z \geq 1.7) \)[/tex], let's consider the properties of the standard normal distribution [tex]\( z \)[/tex].
1. Understanding the Standard Normal Distribution:
- The standard normal distribution is symmetric about the mean, which is [tex]\( 0 \)[/tex].
- Probabilities to the left of a value (e.g., [tex]\( P(z \leq a) \)[/tex]) and to the right of a value (e.g., [tex]\( P(z \geq a) \)[/tex]) such that [tex]\( P(z \leq a) + P(z \geq a) = 1 \)[/tex].
2. Complementary Cumulative Distribution:
- For any given [tex]\( z \)[/tex]-value, the probability that [tex]\( z \)[/tex] is greater than or equal to a certain value is the complement of [tex]\( P(z \leq 1.7) \)[/tex].
- [tex]\( P(z \geq 1.7) = 1 - P(z \leq 1.7) \)[/tex]
3. Checking Each Statement:
- [tex]\( P(z \geq -1.7) \)[/tex]:
This represents the probability that a [tex]\( z \)[/tex]-value is greater than or equal to [tex]\(-1.7\)[/tex]. This is not equivalent to [tex]\( P(z \geq 1.7) \)[/tex] because [tex]\(-1.7\)[/tex] and [tex]\( 1.7 \)[/tex] are not symmetrical counterparts with respect to the area under the curve in standard normal distribution.
- [tex]\( 1 - P(z \geq -1.7) \)[/tex]:
This represents the complementary probability for [tex]\( z \)[/tex] being less than [tex]\(-1.7\)[/tex], i.e., [tex]\( P(z \leq -1.7) \)[/tex]. This also is not equivalent to [tex]\( P(z \geq 1.7) \)[/tex].
- [tex]\( P(z \leq 1.7) \)[/tex]:
This is the cumulative probability that [tex]\( z \)[/tex] is less than or equal to [tex]\( 1.7 \)[/tex]. Given that the cumulative probability up to [tex]\( 1.7 \)[/tex] accounts for everything to the left, the complementary probability would be [tex]\( 1 - P(z \leq 1.7) \)[/tex], which exactly matches [tex]\( P(z \geq 1.7) \)[/tex].
- [tex]\( 1 - P(z \geq 1.7) \)[/tex]:
This expression represents the complementary probability to [tex]\( P(z \geq 1.7) \)[/tex], which is actually [tex]\( P(z \leq 1.7) \)[/tex] - not the probability itself.
4. Result:
- According to the properties of the cumulative distribution function (CDF), to find [tex]\( P(z \geq 1.7) \)[/tex], we can use the complement of [tex]\( P(z \leq 1.7) \)[/tex].
Since [tex]\( P(z \geq 1.7) \)[/tex] is the complement of [tex]\( P(z \leq 1.7) \)[/tex], the statement equivalent to [tex]\( P(z \geq 1.7) \)[/tex] is [tex]\( P(z \leq 1.7) \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{P(z \leq 1.7)} \][/tex]
1. Understanding the Standard Normal Distribution:
- The standard normal distribution is symmetric about the mean, which is [tex]\( 0 \)[/tex].
- Probabilities to the left of a value (e.g., [tex]\( P(z \leq a) \)[/tex]) and to the right of a value (e.g., [tex]\( P(z \geq a) \)[/tex]) such that [tex]\( P(z \leq a) + P(z \geq a) = 1 \)[/tex].
2. Complementary Cumulative Distribution:
- For any given [tex]\( z \)[/tex]-value, the probability that [tex]\( z \)[/tex] is greater than or equal to a certain value is the complement of [tex]\( P(z \leq 1.7) \)[/tex].
- [tex]\( P(z \geq 1.7) = 1 - P(z \leq 1.7) \)[/tex]
3. Checking Each Statement:
- [tex]\( P(z \geq -1.7) \)[/tex]:
This represents the probability that a [tex]\( z \)[/tex]-value is greater than or equal to [tex]\(-1.7\)[/tex]. This is not equivalent to [tex]\( P(z \geq 1.7) \)[/tex] because [tex]\(-1.7\)[/tex] and [tex]\( 1.7 \)[/tex] are not symmetrical counterparts with respect to the area under the curve in standard normal distribution.
- [tex]\( 1 - P(z \geq -1.7) \)[/tex]:
This represents the complementary probability for [tex]\( z \)[/tex] being less than [tex]\(-1.7\)[/tex], i.e., [tex]\( P(z \leq -1.7) \)[/tex]. This also is not equivalent to [tex]\( P(z \geq 1.7) \)[/tex].
- [tex]\( P(z \leq 1.7) \)[/tex]:
This is the cumulative probability that [tex]\( z \)[/tex] is less than or equal to [tex]\( 1.7 \)[/tex]. Given that the cumulative probability up to [tex]\( 1.7 \)[/tex] accounts for everything to the left, the complementary probability would be [tex]\( 1 - P(z \leq 1.7) \)[/tex], which exactly matches [tex]\( P(z \geq 1.7) \)[/tex].
- [tex]\( 1 - P(z \geq 1.7) \)[/tex]:
This expression represents the complementary probability to [tex]\( P(z \geq 1.7) \)[/tex], which is actually [tex]\( P(z \leq 1.7) \)[/tex] - not the probability itself.
4. Result:
- According to the properties of the cumulative distribution function (CDF), to find [tex]\( P(z \geq 1.7) \)[/tex], we can use the complement of [tex]\( P(z \leq 1.7) \)[/tex].
Since [tex]\( P(z \geq 1.7) \)[/tex] is the complement of [tex]\( P(z \leq 1.7) \)[/tex], the statement equivalent to [tex]\( P(z \geq 1.7) \)[/tex] is [tex]\( P(z \leq 1.7) \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{P(z \leq 1.7)} \][/tex]
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