Question

A nutrition laboratory tests 50 ​"reduced sodium" hot​ dogs, finding that the mean sodium content is 321 ​mg, with a standard deviation of 37 mg. ​a) Find a 95​% confidence interval for the mean sodium content of this brand of hot dog. ​b) What assumptions have you made in this​ inference? Are the appropriate conditions​ satisfied? ​c) Explain clearly what your interval means.

Answer 1

a)
sample mean, xbar = 31
sample standard deviation, s = 37
sample size, n = 50
degrees of freedom, df = n - 1 = 49

Given CI level is 95%, hence α = 1 - 0.95 = 0.05
α/2 = 0.05/2 = 0.025, tc = t(α/2, df) = 2.01

CI = (xbar - tc * s/sqrt(n) , xbar + tc * s/sqrt(n))
CI = (31 - 2.01 * 37/sqrt(50) , 31 + 2.01 * 37/sqrt(50))
CI = (20.48 , 41.52)

b)
assumptions:
Random sample is done from a normally distributed population

c)
This interval means one can be 95% confident that the sodium content in such hot dogs will be within 20.48 and 41.52