A nutrition laboratory tests 40 “reduced sodium” hot dogs, finding that the mean
ID: 3217236 • Letter: A
Question
A nutrition laboratory tests 40 “reduced sodium” hot dogs, finding that the mean sodium content is 310 mg, with a standard deviation of 36 mg. a) Find a 95% confidence interval of the mean sodium content of this brand of hot dog. b) What assumptions have you made in this inference? c) Explain what your interval means. The nutrition lab tests hotdogs again, this time using a sample of 60 “reduced sodium” hot dogs. The new sample produces a mean of 318 mg of sodium, and the standard deviation is 32 mg. d) Should this larger sample of hot dogs produce a more accurate estimate of the mean sodium content? Explain. e) What is the standard error of the mean sodium content? Recall that the standard error is the estimate of the standard deviation of the sample mean. f) Find and interpret a 95% confidence interval. g) Food labeling regulations require that any food identified as “reduced sodium” must have at least 30% less sodium than its regular counterpart. If regular franks average 465 mg of sodium, should this brand be labeled “reduced”? Explain, using your confidence interval.
Explanation / Answer
(a)
n = 40
x-bar = 310
s = 36
% = 95
Standard Error, SE = s/n = 36/40 = 5.692099788
Degrees of freedom = n - 1 = 40 -1 = 39
t- score = 2.022690901
Width of the confidence interval = t * SE = 2.02269090124204 * 5.69209978830308 = 11.51335845
Lower Limit of the confidence interval = x-bar - width = 310 - 11.5133584507624 = 298.4866415
Upper Limit of the confidence interval = x-bar + width = 310 + 11.5133584507624 = 321.5133585
The 95% confidence interval is [298.49, 321.51]
(b) We have assumed that the mean sodium content is normally distributed.
(c) We are 95% sure that the true mean sodium content lies in the above interval
(d) Yes, because as the sample size increases, the standard error decreases and the result will be more accurate.
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