write the binomial probability in words. then, use a continuity correction to convert the binomial…
write the binomial probability in words. then, use a continuity correction to convert the binomial probability to a normal distribution probability.\n$p(x < 54)$\nwrite the probability in words.\nthe probability of getting fewer than 54 successes.\nwhich of the following is the normal probability statement that corresponds to the binomial probability statement?\na. $p(x > 53.5)$\nb. $p(x < 53.5)$\nc. $p(x < 54.5)$\nd. $p(53.5 < x < 54.5)$\ne. $p(x > 54.5)$
Answer
Explanation:
Step1: Interpret the binomial probability statement
The expression $P(x < 54)$ for a discrete binomial distribution represents the probability of obtaining a number of successes strictly less than 54.
Step2: Identify the discrete values included
Since $x$ must be an integer, $x < 54$ is equivalent to $x \le 53$. The values included are ${0, 1, 2, \dots, 53}$.
Step3: Apply the continuity correction factor
To approximate a discrete distribution with a continuous one, we expand the interval by $0.5$ units. For the upper bound of the included values (53), we extend to $53 + 0.5$.
Step4: Formulate the normal distribution statement
The continuous interval covering all integers up to 53 ends at $53.5$. Therefore, the normal probability statement is $P(x < 53.5)$.
Answer:
The probability of getting fewer than 54 successes. B. $P(x < 53.5)$