A psychic was tested for ESP. The psychic was presented with 400 cards face down and was asked to determine if the card was one of four symbols: a cross, a star, a circle, or a square. The psychic was correct in 120 of the cases. Let p represent the probability that the psychic correctly identifies the symbol on the card in a random trial.
Using the results above, construct a 99% confidence interval for p.
To construct a confidence interval for the probability p, we can use the formula for confidence intervals for proportions.
The formula for a confidence interval for a proportion is given by:
CI = p̂ ± z * √(p̂(1-p̂)/n)
Where:
- CI is the confidence interval
- p̂ is the sample proportion (in this case, the proportion of correct identifications by the psychic, which is 120/400 = 0.3)
- z is the z-score associated with the desired confidence level (in this case, for a 99% confidence level, z ≈ 2.576)
- n is the sample size (400 in this case)
Plugging in the values, we can calculate the confidence interval as follows:
CI = 0.3 ± 2.576 * √(0.3(1-0.3)/400)
Calculating the values:
CI = 0.3 ± 2.576 * √(0.3 * 0.7 / 400)
CI = 0.3 ± 2.576 * √(0.21 / 400)
CI = 0.3 ± 2.576 * 0.0158
CI = 0.3 ± 0.0407
Therefore, the 99% confidence interval for p is approximately (0.2593, 0.3407).
To construct a confidence interval for the probability p, we can use the formula for proportions.
First, let's determine the sample proportion (p-hat) which represents the proportion of correct responses observed in the sample. In this case, the psychic was correct in 120 out of 400 trials, so the sample proportion is:
p-hat = 120/400 = 0.3
Next, we need to calculate the standard error (SE) of the sample proportion. The formula for standard error is:
SE = sqrt((p-hat * (1 - p-hat)) / n)
where n is the sample size. In this case, the sample size is 400. So we can substitute the values into the formula:
SE = sqrt((0.3 * (1 - 0.3)) / 400)
Now we can calculate the margin of error (ME) using the a critical value from the standard normal distribution corresponding to the desired confidence level. Since we want a 99% confidence interval, the corresponding critical value is 2.576 (approximately). The formula for margin of error is:
ME = critical value * SE
So the margin of error is:
ME = 2.576 * SE
Finally, we can construct the confidence interval by subtracting and adding the margin of error to the sample proportion:
Confidence interval = p-hat ± ME
Substituting the values we calculated earlier:
Confidence interval = 0.3 ± ME
Now we just need to calculate the margin of error and complete the confidence interval.
Use a confidence interval formula for proportions.
Here's one example:
CI99 = p + or - (2.58)(sqrt of pq/n)
...where sqrt = square root, p = x/n, q = 1 - p, and n = sample size.
p = 120/400
q = 280/400
n = 400
Convert all fractions to decimals before using the formula to plug in the numbers.
I'll let you take it from here.