A random sample of 100 cans of peaches has an observed mean weight of 48.5 ounces and a standard deviation of 2.4 ounces. Determine the lower limit of the 95% interval for the true mean weight. The can label says that the can weights 48 ounces. Comment on this claim.
To determine the lower limit of the 95% interval for the true mean weight, we need to use a confidence interval formula that takes into account the sample mean, sample standard deviation, sample size, and the desired level of confidence.
The formula for the lower limit of the confidence interval is:
Lower limit = sample mean - (critical value * (sample standard deviation / sqrt(sample size)))
In this case, the sample mean is 48.5 ounces, the sample standard deviation is 2.4 ounces, and the sample size is 100 cans. The level of confidence is 95%, which corresponds to a critical value.
For a 95% confidence level, the critical value is approximately 1.96.
Plugging in the values into the formula, we have:
Lower limit = 48.5 - (1.96 * (2.4 / sqrt(100)))
Lower limit = 48.5 - (1.96 * (2.4 / 10))
Lower limit = 48.5 - (1.96 * 0.24)
Lower limit = 48.5 - 0.4704
Lower limit ≈ 48.03
Therefore, the lower limit of the 95% interval for the true mean weight is approximately 48.03 ounces.
Now, let's comment on the claim made by the can label, which states that the can weighs 48 ounces. Based on our calculation, the lower limit of the 95% confidence interval for the true mean weight is higher than the claimed weight on the label. This means that it is likely, with 95% confidence, that the true mean weight is higher than what is stated on the can label. So, the claim made by the can label may not be accurate.