An archer normally hits the target with probability of 0.6 ....

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 effective if the probability in part (a) is 0.05 or less.)

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 effective if the probability in part (a) is 0.05 or less.)

Key Concepts

Binomial Distribution

The binomial distribution is a probability distribution that models the number of successes in a fixed number of independent trials, each with the same probability of success. In this problem, it is used to calculate the likelihood of an archer hitting the target a certain number of times out of a given number of attempts, given her probability of success on each shot.

Cumulative Probability

Cumulative probability in the context of the binomial distribution refers to the probability of achieving at least a certain number of successes. This is done by summing the probabilities of all outcomes that meet or exceed the specified threshold. In the given problem, it involves summing the probabilities of hitting the target five, six, seven, or eight times out of eight attempts.

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample of data to infer that a certain condition holds for the entire population. In the context of the problem, it involves assessing whether the observed improvement in the archer's performance after coaching is statistically significant, by comparing the p-value to a predefined significance level.

Statistical Significance

Statistical significance is a measure of whether an observed effect in a study is likely to be genuine or occurred by chance. Typically, a threshold (such as 0.05) is set; if the calculated p-value falls below this threshold, the effect is considered statistically significant. The problem uses this concept to determine whether the new coaching appears to have made an effective difference in the archer's performance.

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