Hypothesis TestingA factory claims that the average lifetime of its light bulbs is 1200 hours. A sample of 25 bulbs has a mean lifetime of 1170 hours with a standard deviation of 80 hours. At the 5% significance level, test whether the claim is true.

The weights of packets of sugar are normally distributed with mean 1.02 kg and standard deviation 0.05 kg.(a) What proportion of packets weigh more than 1.10 kg?(b) What is the probability that a randomly chosen packet weighs between 0.95 and 1.05 kg?

The marks of 6 students in Mathematics are: 35, 40, 50, 60, 70, 80.The marks of the same students in Statistics are: 30, 35, 45, 55, 65, 75.(a) Calculate the correlation coefficient.(b) Find the regression equation of Statistics marks on Mathematics marks.

The average weekly income of a random sample of 50 workers is $540 with a standard deviation of $40.(a) Find the 95% confidence interval for the true mean weekly income.(b) Interpret the result.

A die is rolled once. Let X be the number showing on the die.(a) Write down the probability distribution of X.(b) Find the mean and variance of X.

The lifespans in years of 8 batteries are:4.5, 5.2, 6.0, 5.8, 4.9, 5.5, 6.3, 5.0 Find the mean and standard deviation of the lifespans. Construct a 95% confidence interval for the mean lifespan (assume the data is normally distributed).

The following data gives the marks of 5 students in Maths (X) and Statistics (Y): X: 35, 40, 50, 60, 65Y: 30, 35, 48, 52, 60 (i) Find the correlation coefficient.(ii) Find the regression line of Y on X.

The marks of 12 students are:15, 20, 25, 30, 20, 18, 25, 22, 20, 30, 15, 25. Tasks:a) Find the mean, median, and mode.b) Compute the variance and standard deviation.c) Find the coefficient of variation.

Final Answers:** * Amplitude: ≈ **6.71*** Phase shift: ≈ **26.6°**, representing timing adjustment of the signal* Complex numbers are crucial because they compactly describe amplitude + phase, simplify frequency analysis, and make signal processing mathematically tractable.