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Let the random variable denote the concentration of particulate matter.
We find the probability,
The probability that the mean concentration of particulate matter in a sample of 87 filter papers will be within 47 ppm of the guideline level is 0.9985
Let the random variable denote the amount of rainfall for the year
The sample variance is,
The standard deviation is,
The hypotheses tested are,
The test statistic(t-score) is,
Therefore, the t score for this sample is -2.92
Let the random variable represent body temperature. We are given that,
If we were to perform an hypothesis test for the mean, we would apply the t distribution where the number of degrees of freedom would be
Therefore, the number of degrees of freedom for this test is 19.