Question
For the standard normal distribution, find the value of c such that:
P(z > c) = 0.5309
P(z > c) = 0.5309
Answers
GPT-4 Turbo
To find the value of c for which P(z > c) = 0.5309 in a standard normal distribution, we need to look at the z-tables or use statistical software or a calculator that provides the cumulative distribution function (CDF) for the standard normal distribution.
However, the standard normal distribution is symmetric around 0, so if P(z > c) = 0.5309, it implies that P(z < c) = 1 - 0.5309 = 0.4691, since the total area under the standard normal curve is 1.
By finding the z-value (c) that corresponds to the cumulative probability of 0.4691 to the left of c, we actually find the value of c we are looking for.
Standard z-tables, software, or calculators can provide the z-value directly for this cumulative probability. Let's check the tables or see if we can get a rough idea of the value.
If we look at a z-table, we need to find the closest value to 0.4691 in the body of the table. If you find the value of 0.4691 or the nearest value to it, locate the corresponding row (representing the first two digits and one decimal place of z) and column (representing the second decimal place of z) to determine the z-score.
Once you locate 0.4691 or the closest value in the z-table, note down the corresponding z-score. If you are unable to find the exact value, you would typically interpolate between the two closest values, which represent the probabilities slightly lower and slightly higher than 0.4691.
However, since I don't have access to a z-table right now, I will use software or a calculator to estimate this value:
Using statistical software or a calculator, you can typically determine the z-value through a function like "inverse CDF" or "quantile." For a cumulative probability of 0.4691, the inverse CDF would return the associated z-score.
For a rough estimation, consider that z-values near the center of the distribution are approximately:
- 0.00 for the 50% percentile (the median, with half of the area on either side)
- -0.67 for the 25% percentile
- +0.67 for the 75% percentile
Given that 0.4691 is close to 0.5 (but less than it), we would expect the corresponding z-score to be slightly less than 0 as well. Therefore, the value of c that we are looking for will be slightly negative. A common statistical software would give us the precise number, which would likely be in the range of -0.05 to -0.1. We can use statistical software or a standard normal distribution calculator to get the precise value.
If you have access to statistical software, use the inverse CDF or quantile function and input 0.4691 to get the exact value for c. For example, the R command would be `qnorm(0.4691)` and Python's scipy.stats library would have `norm.ppf(0.4691)`.
However, the standard normal distribution is symmetric around 0, so if P(z > c) = 0.5309, it implies that P(z < c) = 1 - 0.5309 = 0.4691, since the total area under the standard normal curve is 1.
By finding the z-value (c) that corresponds to the cumulative probability of 0.4691 to the left of c, we actually find the value of c we are looking for.
Standard z-tables, software, or calculators can provide the z-value directly for this cumulative probability. Let's check the tables or see if we can get a rough idea of the value.
If we look at a z-table, we need to find the closest value to 0.4691 in the body of the table. If you find the value of 0.4691 or the nearest value to it, locate the corresponding row (representing the first two digits and one decimal place of z) and column (representing the second decimal place of z) to determine the z-score.
Once you locate 0.4691 or the closest value in the z-table, note down the corresponding z-score. If you are unable to find the exact value, you would typically interpolate between the two closest values, which represent the probabilities slightly lower and slightly higher than 0.4691.
However, since I don't have access to a z-table right now, I will use software or a calculator to estimate this value:
Using statistical software or a calculator, you can typically determine the z-value through a function like "inverse CDF" or "quantile." For a cumulative probability of 0.4691, the inverse CDF would return the associated z-score.
For a rough estimation, consider that z-values near the center of the distribution are approximately:
- 0.00 for the 50% percentile (the median, with half of the area on either side)
- -0.67 for the 25% percentile
- +0.67 for the 75% percentile
Given that 0.4691 is close to 0.5 (but less than it), we would expect the corresponding z-score to be slightly less than 0 as well. Therefore, the value of c that we are looking for will be slightly negative. A common statistical software would give us the precise number, which would likely be in the range of -0.05 to -0.1. We can use statistical software or a standard normal distribution calculator to get the precise value.
If you have access to statistical software, use the inverse CDF or quantile function and input 0.4691 to get the exact value for c. For example, the R command would be `qnorm(0.4691)` and Python's scipy.stats library would have `norm.ppf(0.4691)`.