Question

You are a researcher studying a certain species of fish in a local lake. You gather a random sample of 18 fish of this species and record the lengths (in centimeters) of the fish. The data is displayed in the table below. Estimate the mean length of all fish of this type in the lake. Use a 98% confidence level. 10.8 9.4 7.7 10.9 6.8 8.9 8.3 8.5 8.4 8 11.3 7.9 8.6 8.5 11.1 11.3 8.1 6.9 (Checksum: 161.4) a) State the parameter of interest, and verify that the necessary conditions are present in order to carry out the inference procedure. b) Find the critical value and standard error. Critical Value: Standard Error: c) Find the confidence interval: ( , ) d) Interpret your 98% confidence interval in context. e) Interpret the confidence level (not this specific interval). What does it even mean to be 98% confident?

Answer 1

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(a) The parameter of interest is what we are studying. Here it is the length of a certain species of fish in a local lake.

The Assumptions are:

(i) The Sample is a simple random sample.

(ii) The samples are independent of each other.

(iii) The sample comes from a population that is normally distributed or approximately normally distributed.

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(b) The Total number of observation = 19. Since population standard deviation is unknown, we use the students t test.

The Critical value at = (100 - 98) / 100 = 0.02 for df = n - 1 = 18 is 2.552

The Standard error = s / sqrt(n) = 1.4844 / sqrt(19) = 0.3405

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(c) The 98% Confidence Interval for the mean (\sigma unknown)

From the given data: = 8.967 cms and s = 1.4844 cms

The Confidence Interval is given by , where

ME = tcritical * Standard Error = 2.552 * 0.3405 = 0.869

The Lower Limit = 8.967 - 0.869 = 8.098

The Upper Limit = 8.967 + 0.869 = 9.836

The 98% Confidence Interval is (8.098 , 9.836)

[(If required to 2 decimals then (8.10, 9.84)]

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(d) Interpretation: We are 98% confident that the population mean length of fish in the local lake lies within the limits from 8.098 to 9.836.

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(e) In General Terms the confidence level is a probability in terms of a percentage

A 98% confidence interval means that if we were to take 100 different samples and compute a 98% confidence interval for each sample, then approximately 98 of the 100 confidence intervals will contain the true mean value ().

In other words, If we repeated draw samples of the same size from the same population, 98% of the values of the sample means will result in a confidence interval that includes the population mean.

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