Question

A self proclaimed psychic was tested for ESP. The psychic was presented with 200 cards face down and asked to determine if the card was one of five symbols: a star, cross, circle, square, or three wavy lines. The psychic was correct in 46 cases. Let p represent the probability that the psychic correctly identifies the symbol on the card in a random trial. Based on the results of the test, find a 95% confidence interval for p. (Assume the 200 trials can be treated as an SRS from the population of all guesses the psychic would make in her lifetime). What can you conclude about her psychic powers?

Answer 1

The 95% confidence interval would be given (0.172;0.288).

We are confident at 95% that the true probability that the psychic correctly identifies the symbol on the card in a random trial is between (0.172;0.288).

We can conclude that her random guessing would have got her correct 20% of the time. Since the confidence interval contains 0.2, she has no psychic powers, rather she just guessed it like normal people. Any person could have got it correct this many times.

Explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

Description in words of the parameter p

`p` represent the probability that the psychic correctly identifies the symbol on the card in a random trial

`\hat p` represent the estimated probability that the psychic correctly identifies the symbol on the card in a random trial

n=200 is the sample size required

`z_(\alpha/2)` represent the critical value for the margin of error

The population proportion have the following distribution

`p \sim N(p,\sqrt{(p(1-p))/(n)})`

Numerical estimate for p

In order to estimate a proportion we use this formula:

`\hat p =(X)/(n)` where X represent the cases successful

`\hat p=(46)/(200)=0.23` represent the estimated probability that the psychic correctly identifies the symbol on the card in a random trial

Confidence interval

The confidence interval for a proportion is given by this formula:

`\hat p \pm z_(\alpha/2) \sqrt{(\hat p(1-\hat p))/(n)}`

For the 95% confidence interval the value of
`\alpha=1-0.95=0.05` and
`\alpha/2=0.025`, with that value we can find the quantile required for the interval in the normal standard distribution.

`z_(\alpha/2)=1.96`

And replacing into the confidence interval formula we got:

`0.23 - 1.96 \sqrt{(0.23(1-0.23))/(200)}=0.172`

`0.23 + 1.96 \sqrt{(0.23(1-0.23))/(200)}=0.288`

And the 95% confidence interval would be given (0.172;0.288).

We are confident at 95% that the true probability that the psychic correctly identifies the symbol on the card in a random trial is between (0.172;0.288).

We can conclude that her random guessing would have got her correct 20% of the time. Since the confidence interval contains 0.2, she has no psychic powers, rather she just guessed it like normal people. Any person could have got it correct this many times.