Question

to test whether someone has misinatory perception (msp), choose 1 of 4 cards at random--a star, wave, cross, or circle. ask the person to identify the card without seeing it. do this a total of 20 times and see how many cards the person identifies correctly. let x = the number of correct identifications, assuming that the person does not have msp and is just guessing on each card. (a) explain why x is a binomial random variable. x is a binomial random variable because the following conditions are met. binary? independent? number? same probability? (b) calculate the mean of x. mean = cards (do not round.) interpret the mean of x. if 20 cards are selected, we expect about cards to be correctly identified, on average. (c) calculate the standard deviation of x. sd = cards. (round to 2 decimal places.) interpret the standard deviation of x. if 20 cards are selected, the number of correctly identified cards would typically vary from to by about cards.

Explanation:

Part (a) Explanation:

To determine if \( X \) is a binomial random variable, we check the four conditions for a binomial experiment:

1. Binary? Each trial (card identification) has two outcomes: correct or incorrect.
2. Independent? The person is guessing, so each trial (card) is independent of others.
3. Number? There are a fixed number of trials: \( n = 20 \).
4. Same probability? The probability of success (correct identification) is constant for each trial. Since there are 4 cards, \( p = \frac{1}{4} = 0.25 \).

Part (b) Explanation:

The mean of a binomial random variable is given by \( \mu = np \). Here, \( n = 20 \) and \( p = \frac{1}{4} \).

Step 1: Identify \( n \) and \( p \)

\( n = 20 \), \( p = \frac{1}{4} \).

Step 2: Calculate the mean

\( \mu = np = 20 \times \frac{1}{4} = 5 \).

Part (c) Explanation:

The standard deviation of a binomial random variable is \( \sigma = \sqrt{np(1 - p)} \).

Step 1: Identify \( n \), \( p \), and \( 1 - p \)

\( n = 20 \), \( p = \frac{1}{4} \), so \( 1 - p = \frac{3}{4} \).

Step 2: Calculate \( np(1 - p) \)

\( np(1 - p) = 20 \times \frac{1}{4} \times \frac{3}{4} = 20 \times \frac{3}{16} = \frac{60}{16} = 3.75 \).

Step 3: Take the square root

\( \sigma = \sqrt{3.75} \approx 1.94 \) (rounded to two decimal places).

Final Answers:

(a) \( X \) is binomial because:

• Binary: Each trial has two outcomes (correct/incorrect).
• Independent: Trials are independent (guessing).
• Number: Fixed trials (\( n = 20 \)).
• Same probability: \( p = 0.25 \) for each trial.

(b) Mean of \( X \): \( \boldsymbol{5} \) cards.

(c) Standard deviation of \( X \): \( \boldsymbol{1.94} \) cards.

Answer:

Part (a) Explanation:

To determine if \( X \) is a binomial random variable, we check the four conditions for a binomial experiment:

1. Binary? Each trial (card identification) has two outcomes: correct or incorrect.
2. Independent? The person is guessing, so each trial (card) is independent of others.
3. Number? There are a fixed number of trials: \( n = 20 \).
4. Same probability? The probability of success (correct identification) is constant for each trial. Since there are 4 cards, \( p = \frac{1}{4} = 0.25 \).

Part (b) Explanation:

The mean of a binomial random variable is given by \( \mu = np \). Here, \( n = 20 \) and \( p = \frac{1}{4} \).

Step 1: Identify \( n \) and \( p \)

\( n = 20 \), \( p = \frac{1}{4} \).

Step 2: Calculate the mean

\( \mu = np = 20 \times \frac{1}{4} = 5 \).

Part (c) Explanation:

The standard deviation of a binomial random variable is \( \sigma = \sqrt{np(1 - p)} \).

Step 1: Identify \( n \), \( p \), and \( 1 - p \)

\( n = 20 \), \( p = \frac{1}{4} \), so \( 1 - p = \frac{3}{4} \).

Step 2: Calculate \( np(1 - p) \)

\( np(1 - p) = 20 \times \frac{1}{4} \times \frac{3}{4} = 20 \times \frac{3}{16} = \frac{60}{16} = 3.75 \).

Step 3: Take the square root

\( \sigma = \sqrt{3.75} \approx 1.94 \) (rounded to two decimal places).

Final Answers:

(a) \( X \) is binomial because:

• Binary: Each trial has two outcomes (correct/incorrect).
• Independent: Trials are independent (guessing).
• Number: Fixed trials (\( n = 20 \)).
• Same probability: \( p = 0.25 \) for each trial.

(b) Mean of \( X \): \( \boldsymbol{5} \) cards.

(c) Standard deviation of \( X \): \( \boldsymbol{1.94} \) cards.