You intend to estimate a population proportion with a confidence interval. The data suggests that the normal distribution is a reasonable approximation for the binomial distribution in this case.
While it is an uncommon confidence level, find the critical value that corresponds to a confidence level of 94.8%.
(Report answer accurate to three decimal places with appropriate rounding.)
z/2 = ±
If you are looking for the Z score, find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (±.474) to get its Z score.(1.95)
To find the critical value that corresponds to a confidence level of 94.8%, we need to determine the value of zα/2.
The confidence level is given as 94.8%, which means the remaining area under the normal distribution curve is (100% - 94.8%) / 2 = 2.6% on each tail.
To find the zα/2 value, we can use a standard normal distribution table or a calculator.
Using a standard normal distribution table, we can find the value that corresponds to an area of 2.6% in the tail. The table provides the z-score for the cumulative proportion (in the body of the distribution), so we need to subtract from 1 to get the z-score for the tail.
Looking up the value in the table, the closest entry to 2.6% is 0.9949, which corresponds to a z-score of approximately 2.332.
Therefore, the critical value that corresponds to a confidence level of 94.8% is ±2.332 (to three decimal places).
To find the critical value that corresponds to a confidence level of 94.8%, we need to determine the value of zα/2. The confidence level, 94.8%, corresponds to an alpha value (α) of 1 - 94.8% = 5.2%.
Since we're assuming that the normal distribution is a reasonable approximation for the binomial distribution, we can use the standard normal distribution table to find the critical value.
The critical value, zα/2, is the z-score that separates the middle 94.8% of the distribution from the tails.
To find this value:
1. Start by dividing α by 2 to get α/2: α/2 = 5.2% / 2 = 2.6%.
2. Look for the cumulative probability closest to α/2 in the standard normal distribution table. The closest cumulative probability is 0.4950, which corresponds to the z-score of 1.96.
3. Since we're using a two-tailed test (±), the critical value is the negative and positive versions of 1.96.
Therefore, the critical value that corresponds to a confidence level of 94.8% is ±1.96.