Question

In the June 1986 issue of Consumer Reports, some data on the calorie content of beef hot dogs is given. Here are the numbers of calories in 20 different hot dog brands:

186,181,176,149,184,190,158,139,175,148,

152,111,141,153,190,157,131,149,135,132.

Assume that these numbers are the observed values from a random sample of twenty independent standard random variables with meanμand variance \({{\bf{\sigma }}^{\bf{2}}}\), both unknown. Find a 90% confidence interval for the mean number of caloriesμ.

Short answer

The 90% confidence interval for the mean number of calories is(148.09,165.60)

Step-by-step solution

Given information

There is a calorie content dataset of beef hot dogs. The dataset provides the number of calories in 20 different hot dog brands. The dataset is,

The numbers of calories are observed from a random sample of 20 independent normal random variables with mean \(\mu \) and variance \({\sigma ^2}\).

Determine some values from the data

Consider the sample mean \({\bar X_n} = \frac{1}{{20}}\sum\nolimits_{i = 1}^{20} {{X_i}} = 156.85\)

Now the standard deviation is\(\sigma ' = \frac{1}{{19}}\sum\nolimits_{i = 1}^{20} {\left( {{X_i} - {{\bar X}_n}} \right) = 22.64} \)

Now \(T_{19}^{ - 1}\left( {\frac{{1 + 0.90}}{2}} \right) = 1.729\)

Calculate the confidence interval

As there is said that the variance is unknown, so the confidence interval is,

\(\left( {{{\bar X}_n} - T_{n - 1}^{ - 1}\left( {\frac{{1 + \gamma }}{2}} \right)\frac{{\sigma '}}{{\sqrt n }},{{\bar X}_n} + T_{n - 1}^{ - 1}\left( {\frac{{1 + \gamma }}{2}} \right)\frac{{\sigma '}}{{\sqrt n }}} \right) = \left( {148.09,165.60} \right)\)

Therefore, we can conclude that with 90% confidence, the mean calories in a hot dog are between 148.09 and 165.60.