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The solution calculates 95% confidence intervals for a population mean using both the t-distribution (when population variance is unknown) and the z-distribution (when population variance is known). It also discusses how increasing the confidence level results in a wider confidence interval.

The 95% confidence interval for the population mean is [66.04, 69.96], calculated using a sample mean of 68, a known variance of 22, and a sample size of 22. The minimum sample size needed to achieve a maximum confidence interval width of 1 is 338.

The 95% confidence interval for the population mean is approximately [13.984, 14.416]

The provided text details step-by-step calculations for confidence intervals of means and variances, along with sample size determination, under different scenarios (known and unknown population standard deviation). The solutions involve using z-scores, t-scores, and chi-squared values to construct intervals for population parameters.

The 99% confidence interval for the population mean length of steel sheets is [35.71, 38.29] cm.

The Laplace transform of the piecewise function f(t) is calculated using Heaviside step functions and the time-shifting property of Laplace transforms. The result is expressed in terms of the exponential function and the reciprocal of s squared.

The provided text describes functions, relations, domains, ranges, and inverse functions. It includes examples of relations that are not functions, and explains how to determine the domain and range of a function, including a quadratic function and a rational function. It also demonstrates how to find the inverse of a function.

The solution to the system of equations 5m + n = 26 and m - n = 4 is m = 5 and n = 1. Calculating m^n results in 5^1 = 5.

To solve for x in the equation 2x + 5 = 13, subtract 5 from both sides, then divide both sides by 2 to get x = 4.

The expression evaluates to 0 after calculating the numerator and denominator separately and then dividing them.