Question

The amounts (in ounces) of juice in eight randomly selected juice bottles are: 15.8, 15.6, 15.1, 15.2, 15.1, 15.5, 15.9, 15.5. Construct a 97.5% confidence interval for the mean amount of juice in all such bottles. Assume an approximate Normal distribution.

Answer 1

The required 97.5% confidence interval is

`\text {CI} = \bar{x} \pm t_(\alpha/2)((s)/(√(n) ) ) \\\\\text {CI} = 15.5 \pm 2.8412\cdot (0.31)/(√(8) ) \\\\\text {CI} = 15.5 \pm 2.8412\cdot 0.1096\\\\\text {CI} = 15.5 \pm 0.311\\\\\text {CI} = 15.5 - 0.311, \: 15.5 + 0.311\\\\\text {CI} = (15.19, \: 15.81)\\\\`

Therefore, we are 97.5% confident that the actual mean amount of juice in all such bottles is within the range of 15.19 to 15.81 ounces

.

Explanation:

The amounts (in ounces) of juice in eight randomly selected juice bottles are:

15.8, 15.6, 15.1, 15.2, 15.1, 15.5, 15.9, 15.5

Let us first compute the mean and standard deviation of the given data.

Using Excel,

=AVERAGE(number1, number2,....)

The mean is found to be

`\bar{x} = 15.5`

=STDEV(number1, number2,....)

The standard deviation is found to be

`s = 0.31`

The confidence interval for the mean amount of juice in all such bottles is given by

`$ \text {CI} = \bar{x} \pm t_(\alpha/2)((s)/(√(n) ) ) $\\\\`

Where
`\bar{x}` is the sample mean, n is the samplesize, s is the sample standard deviation and
`t_(\alpha/2)` is the t-score corresponding to a 97.5% confidence level.

The t-score corresponding to a 97.5% confidence level is

Significance level = α = 1 - 0.975 = 0.025/2 = 0.0125

Degree of freedom = n - 1 = 8 - 1 = 7

From the t-table at α = 0.0125 and DoF = 7

t-score = 2.8412

So the required 97.5% confidence interval is

`\text {CI} = \bar{x} \pm t_(\alpha/2)((s)/(√(n) ) ) \\\\\text {CI} = 15.5 \pm 2.8412\cdot (0.31)/(√(8) ) \\\\\text {CI} = 15.5 \pm 2.8412\cdot 0.1096\\\\\text {CI} = 15.5 \pm 0.311\\\\\text {CI} = 15.5 - 0.311, \: 15.5 + 0.311\\\\\text {CI} = (15.19, \: 15.81)\\\\`

Therefore, we are 97.5% confident that the actual mean amount of juice in all such bottles is within the range of 15.19 to 15.81 ounces.