Hitting a Target An archer normally hits the target with...

Hitting a Target An archer normally hits the target with probability of $0.6 .$ She hires a new coach for a series of special lessons. After the lessons she hits the target in five out of eight attempts. (a) Find the probability that she would have hit five or more out of the eight attempts before her lessons with the new coach. (b) Did the new coaching appear to make a difference? (Consider the coaching appear to make a difference? $\quad($ a) is 0.05 or less.)

Hitting a Target An archer normally hits the target with
probability of $0.6 .$ She hires a new coach for a series of special
lessons. After the lessons she hits the target in five out of
eight attempts.
(a) Find the probability that she would have hit five or more
out of the eight attempts before her lessons with the new
coach.
(b) Did the new coaching appear to make a difference?
(Consider the coaching appear to make a difference?
$\quad($ a) is 0.05 or less.)

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Key Concepts

Binomial Distribution

This is a discrete probability distribution that models the number of successes in a fixed number of independent Bernoulli trials, where each trial has the same probability of success. It is fundamental when dealing with binary outcomes and helps in quantifying the likelihood of achieving a particular count of successes over a series of attempts.

Cumulative Probability

Cumulative probability refers to the probability that a random variable is less than or equal to a particular value. In the context of the binomial distribution, it is used to compute the probability of obtaining a range of outcomes (for example, five or more successes) by summing up the individual probabilities for each of those outcomes.

Hypothesis Testing and Significance Levels

Hypothesis testing is a statistical process that compares an observed outcome to a null hypothesis, which typically represents a standard or expected probability. The p-value, which arises from this testing, indicates the probability of observing results as extreme as those measured, assuming the null hypothesis is true. If this p-value is less than a predetermined significance level (commonly 0.05), it suggests that the observed difference (such as improved performance) is statistically significant and unlikely to be due to random chance.

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