A food manufacturer samples 7 bags of pretzels off the assembly line and weighs their contents. If the sample mean is 14.2 oz and the sample standard deviation is 0.60 oz., find the 95% confidence interval of the true mean.

Question

A food manufacturer samples 7 bags of pretzels off the assembly line and weighs their contents. If the sample mean is 14.2 oz and the sample standard deviation is 0.60 oz., find the 95% confidence interval of the true mean.

Answer 1

95% = mean ± 1.96 SEm

SEm = SD/√n

Answer 2

To find the 95% confidence interval of the true mean, we can use the formula:

Confidence Interval = sample mean ± (critical value) * (sample standard deviation / √(sample size))

First, we need to find the critical value. For a 95% confidence level, we need to find the z-score at the 97.5th percentile (since the distribution is symmetric), which corresponds to a tail area of 0.025.

Looking up the z-score for a tail area of 0.025, we find it to be approximately 1.96.

Next, we can substitute the given values into the formula:

Confidence Interval = 14.2 ± (1.96) * (0.60 / √7)

Calculating the square root of 7, we get:

Confidence Interval = 14.2 ± (1.96) * (0.60 / 2.64575)

Simplifying the expression inside the parentheses, we get:

Confidence Interval = 14.2 ± (1.96) * 0.22688

Evaluating the multiplication, we get:

Confidence Interval = 14.2 ± 0.44499

Therefore, the 95% confidence interval for the true mean is approximately (13.755, 14.645) ounces.

Answer 3

To find the 95% confidence interval of the true mean, we can use the formula:

Confidence interval = sample mean ± (critical value * standard deviation / √n)

First, we need to find the critical value for a 95% confidence interval. Since the sample size is only 7, we can alternatively use a t-distribution instead of a normal distribution. For a 95% confidence interval with 6 degrees of freedom (7 samples minus 1), the critical value can be found from the t-distribution table or using statistical software. For simplicity, let's assume the critical value is 2.4469.

Next, let's plug the values into the formula:

Confidence interval = 14.2 ± (2.4469 * 0.60 / √7)

Calculating the standard error of the mean (√n) gives us:

√7 ≈ 2.6458

Now we can substitute the values again:

Confidence interval = 14.2 ± (2.4469 * 0.60 / 2.6458)

Simplifying the equation gives us:

Confidence interval = 14.2 ± 0.5568

Finally, we can calculate the confidence interval:

Lower bound = 14.2 - 0.5568 ≈ 13.6432
Upper bound = 14.2 + 0.5568 ≈ 14.7568

The 95% confidence interval of the true mean is approximately 13.6432 oz to 14.7568 oz.