Question
For a standard normal distribution, find:
P(z < 2.57)
P(z < 2.57)
Answers
Answered by
GPT-4 Turbo
To find the probability that \( z \) is less than 2.57 in a standard normal distribution, you would refer to a standard normal distribution table (Z-table), use a statistical calculator, or a software that can compute such probabilities.
However, if neither is available, one can approximate using the empirical rule, which states that about 95% of the data falls within 2 standard deviations of the mean in a normal distribution. Since 2.57 is more than 2 standard deviations from the mean (0 for a standard normal distribution), the probability would be slightly higher than 95%.
Thankfully, I can provide a pretty accurate approximation using z-scores, where the Z-table or statistical functions in software give more precise results.
For the standard normal distribution (mean = 0, standard deviation = 1):
\( P(z < 2.57) \)
Approximately equals to 0.995 or 99.5%.
You can validate this by looking at a standard normal distribution table or using statistical software or a calculator that includes cumulative distribution function (CDF) capabilities. Specifically, you would look up the value of 2.57 in the Z-table or use the CDF function for the standard normal distribution:
For example, in statistical software like R, you could use:
```R
pnorm(2.57)
```
This would give you a more precise probability value for \( P(z < 2.57) \) in a standard normal distribution. I don't have access to such tools right now, but you can expect it to be very close to or slightly higher than 99.5%.
However, if neither is available, one can approximate using the empirical rule, which states that about 95% of the data falls within 2 standard deviations of the mean in a normal distribution. Since 2.57 is more than 2 standard deviations from the mean (0 for a standard normal distribution), the probability would be slightly higher than 95%.
Thankfully, I can provide a pretty accurate approximation using z-scores, where the Z-table or statistical functions in software give more precise results.
For the standard normal distribution (mean = 0, standard deviation = 1):
\( P(z < 2.57) \)
Approximately equals to 0.995 or 99.5%.
You can validate this by looking at a standard normal distribution table or using statistical software or a calculator that includes cumulative distribution function (CDF) capabilities. Specifically, you would look up the value of 2.57 in the Z-table or use the CDF function for the standard normal distribution:
For example, in statistical software like R, you could use:
```R
pnorm(2.57)
```
This would give you a more precise probability value for \( P(z < 2.57) \) in a standard normal distribution. I don't have access to such tools right now, but you can expect it to be very close to or slightly higher than 99.5%.