1. The amount of pyridoxine (in grams) per multiple vitamin is normally distributed with = 110 grams and = 25 grams. A sample of 25 vitamins is to be selected. So, 95% of all sample means will be greater than how many grams?
To find the value of grams at which 95% of all sample means will be greater, we can use the formula for the confidence interval of a sample mean:
Sample Mean = Population Mean ± (Z * (Population Standard Deviation / √n))
Where:
- Sample Mean is the mean value of the sample
- Population Mean is the mean value of the population
- Z is the z-score corresponding to the desired confidence level (in this case, 95% confidence level)
- Population Standard Deviation is the standard deviation of the population
- n is the sample size
Given:
Population Mean (μ) = 110 grams
Population Standard Deviation (σ) = 25 grams
Sample Size (n) = 25
To find the z-score corresponding to a 95% confidence level (α = 0.05), we can use a standard normal distribution table or a calculator. The z-score for a 95% confidence level is approximately 1.96.
Using the formula, we can calculate the margin of error:
Margin of Error = Z * (σ / √n)
= 1.96 * (25 / √25)
= 1.96 * (25 / 5)
= 1.96 * 5
= 9.8 grams
To find the lower limit of the confidence interval, we subtract the margin of error from the population mean:
Lower Limit = Population Mean - Margin of Error
= 110 grams - 9.8 grams
= 100.2 grams
Therefore, 95% of all sample means will be greater than 100.2 grams.
To find the grams value that 95% of all sample means will be greater than, we can use the properties of normal distribution and the formula for the confidence interval.
1. First, let's understand the properties of the normal distribution. The normal distribution is symmetric and bell-shaped, and it is defined by two parameters: the mean (μ) and the standard deviation (σ).
2. In this case, we are given that the mean (μ) of the pyridoxine content is 110 grams, and the standard deviation (σ) is 25 grams.
3. We are interested in the sample mean, so we need to use the formula for the confidence interval. The formula for the confidence interval for the mean is:
Confidence Interval = sample mean ± (z * standard deviation/sqrt(sample size))
Here, z represents the z-value corresponding to the desired confidence level. In this case, we want to find the value for a 95% confidence level. The z-value corresponding to a 95% confidence level is 1.96.
4. Now we can substitute the values into the formula:
Confidence Interval = 110 grams ± (1.96 * 25 grams/sqrt(25))
Simplifying further:
Confidence Interval = 110 grams ± (1.96 * 25 grams/5)
Confidence Interval = 110 grams ± 9.8 grams
5. To find the value that 95% of all sample means will be greater than, we need to add the confidence interval to the sample mean:
110 grams + 9.8 grams = 119.8 grams
Therefore, 95% of all sample means will be greater than approximately 119.8 grams.
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the Z score related to that proportion.
Insert that value into this equation.
Z = (score-mean)/SD
Solve for the value of the score in grams.