Daily water intake (including water used in drinks such as coffee, tea and juice) for Canadian adults follows a normal distribution with mean 1.5 litres and standard deviation 0.31 litres.
(d) Using the 68-95-99.7 Rule, approximately 95% of samples of 22 Canadian adults will have mean daily water intakes between and .
(e) If we take a random sample of 11 Canadian adults, there is an 11% chance that their mean daily water intake will be greater than
I've tried these multiple times and seem to be missing a step, any help would be much appreciated.
for question D since 95% is 2 s.d away i multiplied 0.31 by 2 and added it to the mean then subtracted it from the mean to get two values on either side of the mean but i'm unsure how to use the n=22 information
for question E i attempted to approach it as a revers look up problem (turning 11% into 0.11 and looking up its Z score, then plugging it into the standardizing equation, but this isn't working.)
D) since this is average question with given sample size, you need to use (mean, sigma/root sample size).
This is normal distribution, so you don't need to worry about sample size being greater than 30.
Therefore, you should do = mean - 2 (0.31/squre root 22) and mean + 2(0.31/square root 22)
Then you will get the answer.
Let's break down the problems step by step to help you solve them.
(d) For this question, you are trying to find the range within which the mean daily water intakes of 95% of samples of 22 Canadian adults will fall.
First, let's use the 68-95-99.7 Rule, also known as the empirical rule, which states that for data following a normal distribution:
- Approximately 68% of the data falls within one standard deviation of the mean.
- Approximately 95% of the data falls within two standard deviations of the mean.
- Approximately 99.7% of the data falls within three standard deviations of the mean.
Given that the mean daily water intake is 1.5 liters and the standard deviation is 0.31 liters, we can apply the rule as follows:
- Within two standard deviations of the mean, approximately 95% of the data will fall.
- Lower bound: Mean - (2 * standard deviation)
- Upper bound: Mean + (2 * standard deviation)
To calculate the range, we substitute the given values:
- Lower bound: 1.5 - (2 * 0.31) = 0.88 liters
- Upper bound: 1.5 + (2 * 0.31) = 2.12 liters
Since we are asked to find the range for samples of 22 Canadian adults, this range applies to the means of those samples.
(e) For this question, you need to find the probability that the mean daily water intake of a sample of 11 Canadian adults will be greater than a specific value.
To solve this, we need to use the properties of the normal distribution and standardization.
First, calculate the standard deviation of the sample mean using the formula:
- Standard deviation of the sample mean = standard deviation / sqrt(sample size)
- Standard deviation: 0.31 liters
- Sample size: 11
Standard deviation of the sample mean = 0.31 / sqrt(11) ≈ 0.0934 liters
Now, we can convert the given probability to a z-score (standard score) using the standard normal distribution table, or a calculator.
- The z-score indicates how many standard deviations away from the mean our value of interest is.
- In this case, we want to find the z-score for an 11% probability (or 0.11).
Plug in the known values into the formula:
z = (x - mean) / standard deviation
Rearrange the formula to solve for x (the value of interest):
x = (z * standard deviation) + mean
Substitute the known values to find the value of interest (x):
x = (z * 0.0934) + 1.5
Now, you have the value of x which represents the mean daily water intake at which there is an 11% chance of the mean daily water intake of 11 Canadian adults being greater.
Note: Make sure to check the correct tail of the normal distribution table and use the corresponding z-score. Since we want the probability of the mean being greater than the given value, use the upper tail of the standard normal distribution when looking up the z-score.
I hope this helps you find the solutions to these problems! Let me know if you have any further questions or clarifications.