(a) Suppose we construct a 99% confidence interval. What are we 99% confident ab
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Question
(a) Suppose we construct a 99% confidence interval. What are we 99% confident about? (b) Which of the confidence intervals is wider, 90% or 99%? (c) In computing a confidence interval, when do you use the t-distribution and when do you use z, with normal approximation? (d) How does the sample size affect the width of a confidence interval? Suppose X is a random sample of size n = 1 from a uniform distribution defined on the interval (0, Theta). Construct a 98% confidence interval for theta and interpret. Consider the probability statement P(-2.81 lessthanorequalto Z = X- - mu/sigma/squareroot n lessthanorequalto 2.75) = Kappa where X- is the mean of a random sample of size n from N(Mu, sigma^2) distribution with known sigma^2. (a) Find Kappa. (b) Use this statement to find a confidence interval for Mu. (c) What is the confidence level of this confidence interval? (d) Find a symmetric confidence interval for Mu. A random sample of size 50 from a particular brand of 16-ounce tea packets produced a mean weight of 15.65 ounces. Assume that the weights of these brands of tea packets are normally distributed with standard deviation of 0.59 ounce. Find a 95% confidence interval for the true mean Mu. Let X_1, ellipsis, X_n be a random sample from an N(Mu, sigma^2), where the value of sigma^2 is unknown. (a) Construct a(1-alpha) 100% confidence interval for sigma^2, choosing an appropriate pivot. Interpret its meaning. (b) Suppose a random sample from a normal distribution gives the following summary statistics: n = 21, x- = 44.3, and s = 3.96Explanation / Answer
5.4.1
a) A confidence interval for a parameter is an interval of numbers within which we expect the true value of the population parameter to be contained. When we construct a 99% confidence interval, we are 99% confident that the true value of the parameter is in our confidence interval.
b) The larger the confidence, the wider the interval. 99% is wider than 90%.
c) If population standard deviation is not known, then using t-distribution is correct.
If population standard deviation is known, then using the normal distribution is correct.
d) Increasing the sample size decreases the width of confidence intervals, because it decreases the standard error.
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