A filing machine designed to fill soda bottles with 16 ouncesof soda. The distribution for the weight of the bottles is normal. Twenty bottles are selected and weighted. The sample mean is 15.3 ounces and the standard deviation is 1.5 ounces . Develop a 90% confidence interval for this sample.
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (.05 each direction) and its Z score = 1.645.
90% = mean ± 1.645 SEm
SEm = SD/√n
To develop a 90% confidence interval for the sample mean, we can use the formula:
Confidence Interval = Sample Mean ± (Critical Value * Standard Error)
First, let's calculate the standard error, which is the standard deviation divided by the square root of the sample size:
Standard Error = Standard Deviation / √(Sample Size)
In this case, the standard deviation is 1.5 ounces, and the sample size is 20. Therefore:
Standard Error = 1.5 / √(20) = 1.5 / 4.47 ≈ 0.335 ounces
Next, we need to find the critical value associated with a 90% confidence level. Since the distribution is assumed to be normal, we can use the Z-table to find the critical value.
The Z-table gives us the area under the normal distribution curve and is typically used for confidence intervals. For a 90% confidence interval, we need to find the Z-score that corresponds to an area of 0.05 in each tail (0.05 + 0.05 = 0.10).
Looking up this value in the Z-table, we find that the Z-score is approximately 1.645.
Now we can calculate the confidence interval using the formula:
Confidence Interval = Sample Mean ± (Critical Value * Standard Error)
Confidence Interval = 15.3 ± (1.645 * 0.335)
Confidence Interval = 15.3 ± 0.551
Confidence Interval = (14.749, 15.851)
Therefore, the 90% confidence interval for this sample is (14.749, 15.851) ounces. This means that we can estimate with 90% confidence that the true population mean falls within this interval.