Five independent samples, each of size n, are to be drawn from a normal distribution where sigma is known. For each sample, the interval (y-bar - 0.96 * sigma/sqrt(n), y-bar + 1.06 *sigma/sqrt(n)) will be constructed. What is the probability that at least four of the intervals will contain the unknown mu?
You take a trip by air that involves three independent flights. If there is an 68 percent chance each specific leg of the trip is on time, what is the probability all three flights arrive on time?
To determine the probability that at least four of the intervals will contain the unknown mu, we need to consider the properties of the student's t-distribution and confidence intervals.
First, let's understand the concept of a confidence interval. In statistics, a confidence interval is an estimate of an unknown population parameter (such as the mean, mu) based on the sample data. The interval provides a range of plausible values for the parameter, along with a degree of confidence for this range.
In this case, each sample will have a confidence interval constructed using the formula (y-bar - 0.96 * sigma/sqrt(n), y-bar + 1.06 *sigma/sqrt(n)). Here, y-bar represents the sample mean, sigma is the known standard deviation, and n is the sample size.
The value 0.96 is the critical value associated with a 95% confidence level (we'll discuss this further below), and 1.06 is the corresponding critical value associated with an unknown confidence level. These critical values are used to estimate the appropriate range for the population mean.
Now let's address the concept of the student's t-distribution. When the population standard deviation is unknown (as stated in this question), we use the student's t-distribution instead of the standard normal distribution.
The student's t-distribution takes into account the sample size (n - 1 degrees of freedom) and is used to calculate critical values for constructing confidence intervals when the population standard deviation is unknown.
To calculate the probability that at least four out of the five intervals will contain the unknown mu, we need to use the t-distribution to find the individual probabilities for each interval and then apply the concept of independent events.
The t-distribution assigns probabilities to different ranges of t-values, known as "t-scores." In this case, we want to calculate the probability that each interval includes the population mean mu. This probability corresponds to the probability that the t-score falls inside critical values (0.96 and -1.06, respectively) for the t-distribution with n - 1 degrees of freedom.
By using a t-table or a statistical software, we can find the probabilities associated with these t-scores.
The next step is to assume that the intervals are independent events. This means that the outcomes of one interval (whether it includes mu or not) do not affect the outcomes of the other intervals. With that assumption, we can multiply the individual probabilities of each interval together to find the combined probability of at least four intervals containing mu.
For example, if we define "Success" as an interval containing mu and "Failure" as an interval not containing mu, we can calculate the probability using combinations of successes and failures.
The formula to calculate the probability is as follows:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
- P(X = k) is the probability of achieving k successes
- C(n, k) is the combination formula for selecting k successes from n trials
- p is the probability of success on each trial
- (1-p) is the probability of failure on each trial
Using this formula, you can calculate the probability of at least four intervals containing the unknown mu.