Answer :
Final answer:
The average number of pounds of beef consumed each year per person in the United States, based on the given sample data, is approximately 120 pounds. The sample standard deviation is approximately 6.52 pounds. The 98% confidence interval for the average pounds of beef consumed per person in the United States is approximately 115.2 to 124.8 pounds.
Explanation:
To compute the sample mean, we need to sum up all the data points and divide the sum by the number of data points. In this case, we have a sample of 16 Americans and their respective pounds of beef consumed per year:
118, 115, 125, 110, 112, 130, 117, 112, 115, 120, 113, 118, 119, 122, 123, 126
Adding up all these values, we get a sum of 1920. Dividing this sum by 16 (the number of data points), we find that the sample mean is 120 pounds.
To compute the sample standard deviation, we need to calculate the variance first. The variance is the average of the squared differences between each data point and the mean. Subtracting the mean from each data point, squaring the result, and summing up all these squared differences, we find a sum of 640. Dividing this sum by 15 (the number of degrees of freedom), we get a variance of 42.67. Taking the square root of the variance, we find that the sample standard deviation is approximately 6.52 pounds.
To construct a 98% confidence interval using the sample data, we need to calculate the margin of error and the bounds. The margin of error is calculated by multiplying the standard error (which is the standard deviation divided by the square root of the sample size) by a critical value from the t-distribution. The critical value for a 98% confidence level with 15 degrees of freedom is approximately 2.947. The standard error is calculated by dividing the sample standard deviation by the square root of the sample size. In this case, the standard error is approximately 1.63 pounds. Multiplying the standard error by the critical value, we find that the margin of error is approximately 4.80 pounds. To find the lower bound of the confidence interval, we subtract the margin of error from the sample mean. To find the upper bound, we add the margin of error to the sample mean. Therefore, the 98% confidence interval for the average pounds of beef consumed per person in the United States is approximately 115.2 to 124.8 pounds.
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