Discover answers to your most pressing questions at Westonci.ca, the ultimate Q&A platform that connects you with expert solutions. Get accurate and detailed answers to your questions from a dedicated community of experts on our Q&A platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
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]
Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.
