91Ó°ÊÓ

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 \({\sigma ^2}\).

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

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.