Question

1. A new brand of hot dog claims to have a lower sodium content than the leading brand.

a. A random sample of ten of these new hot dogs results in the following sodium measurements (in milligrams).
370 326 322 297 326 289 293 264 327 331
Estimate the population mean sodium content of this new brand of hot dog based on the ten sampled measurements.
b. Calculate the margin of error associated with your estimate of the population mean from part (a). Round your answer to three decimal places.
c. The mean sodium content of the leading brand of hot dogs is known to be 350 mg. Based on the sample mean and the value of the margin of error for the new brand, is a mean
sodium content of 350 mg a plausible value for the mean sodium content of the new brand? Comment on whether you think the new brand of hot dog has a lower sodium content on
average than the leading brand.
d. Another random sample of 40 new-brand hot dogs is taken. Should this larger sample of hot dogs produce a more accurate estimate of the population mean sodium content than the
sample of size 10? Explain your answer by appealing to the formula for margin of error.

Answer 1

(a) Population mean = 314.5mg

(b) Margin of error = 21.281mg

(c) No, 350mg is not a plausible value

The new brand of hot dog has a lower sodium content on average than the leading brand

(d) Yes

Explanation:

(a) Population mean = (370+326+322+297+326+289+293+264+327+331)/10 = 3145/10 = 314.5mg

(b) Margin of error = t×sd/√n

sd = 29.73, n = 10, degree of freedom = n-1 = 10-1 = 9

Assuming 95% confidence level

t-value corresponding to 9 degrees of freedom and 95% confidence level is 2.262

Margin of error = 2.262×29.73/√10 = 21.281mg

(c) No

Lower limit of new brand = 314.5 - 21.281 = 293.219mg

Upper limit = 314.5 + 21.281 = 335.781mg

The range of sodium content for the new brand of hot dog is between 293.219mg and 335.781mg which is lower than the leading brand (350mg)

(d) Margin of error = t×sd/√n

From the formula above, margin of error varies inversely as the square root of sample size

The smaller the sample size (n = 10) the larger the margin of error and the less accurate is the population mean

The larger the sample size (n = 40) the smaller the margin of error and the more accurate is the population mean