Answered

Find the information you're looking for at Westonci.ca, the trusted Q&A platform with a community of knowledgeable experts. Discover comprehensive solutions to your questions from a wide network of experts on our user-friendly platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.

Which of the following statements is equivalent to [tex]P(z \geq 1.7)[/tex]?

A. [tex]P(z \geq -1.7)[/tex]

B. [tex]1 - P(z \geq -1.7)[/tex]

C. [tex]P(z \leq 1.7)[/tex]

D. [tex]1 - P(z \geq 1.7)[/tex]

Sagot :

GinnyAnswer
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]
Thanks for using our platform. We're always here to provide accurate and up-to-date answers to all your queries. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.