Question - Calculating 95% Confidence Interval for Mean Weight of Pretzel Bags
mathotherconfidence interval calculationmean weight estimationt-distribution calculationsample size and standard deviation
Solution:
To find the 95% confidence interval for the mean weight of the pretzel bags, we can use the formula for the confidence interval for a mean when the population standard deviation is not known and the sample size is small (n < 30). Since we're dealing with a small sample size (n = 7), we'll use the t-distribution:Confidence Interval (CI) = $$\bar{x} \pm t_{\alpha/2} \cdot \frac{s}{\sqrt{n}}$$where:- $$\bar{x}$$ is the sample mean,- $$t_{\alpha/2}$$ is the t-value that corresponds to the $$ \frac{\alpha}{2} $$ percentile of the t-distribution with $$n - 1$$ degrees of freedom,- $$s$$ is the sample standard deviation, and- $$n$$ is the sample size.Given that:- The sample mean, $$\bar{x}$$, is 15.2 oz,- The sample standard deviation, $$s$$, is 0.7 oz, and- The sample size, $$n$$, is 7.We need to find the t-value for a 95% confidence interval and 6 degrees of freedom (since $$n - 1 = 7 - 1 = 6$$). Typically, you would use a t-distribution table or statistical software to find this value. For a 95% confidence interval, we look up the value $$t_{0.025}$$ since the two tails (alpha/2) of the curve each have 2.5% of the area. Assuming a t-value of around 2.447 (you'll want to confirm this value using a t-distribution table or software), we can then calculate the confidence interval:CI = $$15.2 \pm 2.447 \cdot \frac{0.7}{\sqrt{7}}$$First calculate the margin of error ($$ME$$):$$ME = 2.447 \cdot \frac{0.7}{\sqrt{7}}$$$$ME = 2.447 \cdot \frac{0.7}{2.64575}$$$$ME = 2.447 \cdot 0.2642$$$$ME ≈ 0.6465$$Now calculate the confidence interval bounds:Lower bound = $$15.2 - 0.6465$$Lower bound ≈ $$14.5535$$Upper bound = $$15.2 + 0.6465$$Upper bound ≈ $$15.8465$$The 95% confidence interval is therefore approximately (14.5535, 15.8465).Looking at the options provided:a) 14.6 to 15.8b) 13.4 to 17.0c) 14.7 to 15.7d) 14.4 to 16.0The confidence interval we calculated is approximately 14.55 to 15.85, which makes option a) 14.6 to 15.8 the closest to our calculated interval and is thus the correct answer.
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