Sorry for all the questions, I really need help and an explanation so I can fully understand. I really appreciate everything. Thank you.
In a random sample of 519 judges, it was found that 285 had an introvert personality type. Construct a 95% confidence interval for the actual percentage of judges who are introverts
1.Enter the SD.Round your answer to 3 decimal places.
2.Enter the SE for the percentage. Round your answer to two decimal places.
3.What is the z-value for a 95% confidence interval?
4.What is the lower boundary of the interval? Round your answer to one decimal place.
5.What is the upper boundary of the interval? Round your answer to one decimal place.
No problem at all! I'm here to help. Let's break down the steps to construct a 95% confidence interval for the actual percentage of judges who are introverts:
1. To start, we need to calculate the standard deviation (SD). The formula for the standard deviation of a proportion is:
SD = √[(p * (1 - p)) / n]
where:
- p is the proportion of judges who are introverts (285/519 = 0.549)
- n is the sample size (519 in this case)
Therefore, SD = √[(0.549 * (1 - 0.549)) / 519]
Calculate the SD and round your answer to three decimal places.
2. Next, we need to find the standard error (SE) for the percentage. The formula for the standard error is:
SE = SD / √n
where SD is the standard deviation and n is the sample size.
Calculate the SE and round your answer to two decimal places.
3. The z-value for a 95% confidence interval can be looked up on a standard normal distribution table. A 95% confidence interval corresponds to a z-value of 1.96. Note that this value is constant for a 95% confidence interval.
4. To find the lower boundary of the interval, subtract the z-value multiplied by the standard error from the sample proportion (p).
Lower boundary = p - (z-value * SE)
Calculate the lower boundary and round your answer to one decimal place.
5. To find the upper boundary of the interval, add the z-value multiplied by the standard error to the sample proportion (p).
Upper boundary = p + (z-value * SE)
Calculate the upper boundary and round your answer to one decimal place.
I hope this helps! Let me know if you have any further questions.
1. To construct a confidence interval, we need to calculate the standard deviation (SD). The formula to calculate the standard deviation of a proportion is:
SD = sqrt( (p * (1 - p)) / n )
where p is the sample proportion and n is the sample size. In this case, the sample proportion of judges with an introvert personality type is 285/519 = 0.549. Plugging these values into the formula:
SD = sqrt( (0.549 * (1 - 0.549)) / 519 )
After performing the calculation, round the answer to 3 decimal places.
2. The standard error (SE) for the percentage is calculated using the formula:
SE = sqrt( (p * (1 - p)) / n ) * 100%
where p is the sample proportion and n is the sample size. Plugging the values from the previous step:
SE = sqrt( (0.549 * (1 - 0.549)) / 519 ) * 100%
After performing the calculation, round the answer to two decimal places.
3. The z-value for a 95% confidence interval can be found in a standard normal distribution table. For a 95% confidence level, we need to find the z-value that corresponds to a 2.5% probability in the tail of the distribution (since it is a two-tailed test). The z-value for a 2.5% probability is approximately 1.96.
4. To calculate the lower boundary of the interval, we use the formula:
Lower boundary = p - (z * SE)
where p is the sample proportion, z is the z-value, and SE is the standard error. Plugging in the values:
Lower boundary = 0.549 - (1.96 * SE)
After performing the calculation, round the answer to one decimal place.
5. To calculate the upper boundary of the interval, we use the formula:
Upper boundary = p + (z * SE)
where p is the sample proportion, z is the z-value, and SE is the standard error. Plugging in the values:
Upper boundary = 0.549 + (1.96 * SE)
After performing the calculation, round the answer to one decimal place.