Question

A nutrition lab tested 40 randomly selected hot dogs to see if their mean sodium content was less than 325mg upper limit set by regulations for "reduced sodium" franks. The sample yielded a mean of 322 mg with a standard deviation of 11.5 mg.

A) To construct a confidence interval, would you use a z-chart or a t-chart? why?

B) Construct a 90% confidence interval for for estimating the mean sodium content for "reduced sodium" hot dogs. Interpret the confidence interval in a sentence.

C) Test the claim that the mean sodium level for the "reduced sodium" hot dogs is less than the limit of 325mg. Use a significance level of 0.05.

D) Does the confidence interval support the conclusion of the hypothesis test? Explain.

Answer 1

(A) As a thumb rule we use t critical values when

(i) The sample size n is < 30 and the population standard deviation is known and it is not mentioned that the samples are normally or approximately distributed (the population will be normally or approximately normally distributed).

(ii) The sample size is n is > 30, but population standard deviation is unknown.

Therefore in this case although n > 30, the population standard deviation is unknown and we use t critical values.

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(B) 90% Confidence interval for the reduced sodium hot dogs:

Given: = 322 mg, s = 11.5 mg, n = 40

tcritical (2 tail) for = 0.10, for df = n -1 = 39, is 1.6849

The Confidence Interval is given by ME, where

The Lower Limit = 322 - 3.06 = 318.94

The Upper Limit = 322 + 3.06 = 325.06

The 95% Confidence Interval is (318.94 , 325.06)

We are 90% confident that the true population mean of reduced sodium in hot dogs lies between 318.94 to 325.06.

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(C) Left Tailed t test, Single Mean

Given: = 325 mg, = 322 mg, s = 11.25 mg, n = 40, = 0.05

The Hypothesis:

H0: = 325

Ha: < 325

This is a Left tailed test

The Test Statistic: The test statistic is given by the equation:

t observed = -1.65

The p Value: The p value (Left tailed) for t = -1.65, for degrees of freedom (df) = n-1 = 39, is; p value = 0.0535

The Critical Value: The critical value (Left Tail) at = 0.05, for df = 21, tcritical= -1.685

The Decision Rule:  

The Critical Value Method: If tobserved is < -tcritical.

The p-value Method: If P value is < , Then Reject H0.

The Decision:

The Critical Value Method: Since tobserved (-1.65) is > -t critical (-1.685), we Fail to Reject H0.

The p-value Method: Since P value (0.0535) is > (0.05) , We fail to Reject H0.

The Conclusion: There is insufficient evidence at the 95% significance level to conclude that the mean sodium content in hot dogs is less than 325.

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(D) The confidence interval contains 325 within the 2 limits, and therefore there will always be the possibility of the true population mean being equal to 325, and hence we would fail to reject the null hypothesis, which is what we have done in the hypothesis test. Yes, the CI supports the findings.

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