Suppose that a laboratory analyzes biological specimens to determine the concentration of toxins. The laboratory analyzes each specimen five times and reports the mean results. The five measurements for a given specimen are not all equal, but the results follow a normal distribution almost exactly. Assume the mean of the population of all repeated measurements, ?μ, is the true concentration in the specimen. The standard deviation of this distribution is known to be 0.1750 ppm. One of the laboratory’s technicians evaluates the claim that the concentration of toxins in a single specimen is at most 4.300 ppm using a one-sample ?z‑test for a mean. The true concentration is the mean, ?μ, of the population of repeated analyses. ?0:??1:?=4.300 ppm>4.300 ppmH0:μ=4.300 ppmH1:μ>4.300 ppm The technician conducts his test at a significance level of 1%. First, determine the minimum value of the sample mean, ?¯x¯, for which the technician will reject the null hypothesis. You may find a table of critical values from the standard normal distribution or some software manuals helpful. Round your answer to three decimal places. What is the power of the technician's test to reject the null hypothesis if the mean concentration is really ?=4.550μ=4.550 ppm? Give your answer precise to at least two decimal places.
Inverse Normal Distribution
The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.
Mean, Median, Mode
It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.
Z-Scores
A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.
Suppose that a laboratory analyzes biological specimens to determine the concentration of toxins. The laboratory analyzes each specimen five times and reports the
One of the laboratory’s technicians evaluates the claim that the concentration of toxins in a single specimen is at most 4.300 ppm using a one-sample ?z‑test for a mean. The true concentration is the mean, ?μ, of the population of repeated analyses.
The technician conducts his test at a significance level of 1%. First, determine the minimum value of the sample mean, ?¯x¯, for which the technician will reject the null hypothesis. You may find a table of critical values from the standard normal distribution or some software manuals helpful.
Round your answer to three decimal places.
What is the power of the technician's test to reject the null hypothesis if the mean concentration is really ?=4.550μ=4.550 ppm? Give your answer precise to at least two decimal places.
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