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Sampling Distribution of a Sample Mean
Problem #1
“Miller Lite has 1/3 less calories than regular beer” or so the commercial says.
According to Miller, a 12 oz Miller Lite has 96 calories (www.millerlite.com). Suppose that the
actual calorie content varies from bottle to bottle, with a mean of 96 calories and a standard
deviation of 5 calories.
We will sample 48 bottles of Miller Lite, carefully measuring the calories in each bottle, and
report the average number of calories per bottle for the sample.
(a) What are the values of m x and s x ?
(b) Can we say a normal distribution is appropriate for this x ?
(c) What is the probability that such a sample has sample mean less than 95 calories?
(d) What is the probability that such a sample has sample mean between 95.4 calories and 97
calories?
(e) What number of calories represents the highest 2% of all such sample means?
(f) If the sample size were increased to some value larger than 48, how would the distribution of
this sample mean be affected (center, shape, spread)?
Answers
(a) m x = 96 calories and s x =
5
» 0.7217 calories
48
(b) Since the sample size is “large” (48 > 30), the Central Limit Theorem says we can assume a
normal distribution for x .
(c) normalcdf (-¥, 95, 96, 0.7217) = 0.0829
(d) normalcdf ( 95.4, 97, 96, 0.7217) = 0.7142
(e) invNorm ( 0.98, 96, 0.7217) = 97.48 calories and above
(f) Center: m x is not affected by the sample size, so it would remain 96 calories,
Shape: Since the sample size is still larger than 30, we could still assume a normal shape
Spread: s x gets smaller as the sample size increases.