Question

In the wester

United States, water is sometimes diverted from mountain streams and rivers to be used for agriculture and municipal water supplies, Diverting too much water raises the water temperature in the
mountain streams and rivers, affecting the fish that live there. A certain mountain river is abundant in brown trout and has water diverted from it for city use. Assume that the water temperature along, all locations of the distribution with mean 18°C and standard deviation of 2°C
A sport-fishing group plans to measure the water temperature at 64 randomly selected locations along the river, Let the random variable F represent the mean water temperature, in degrees Celsius, of 64 locations along the river selected at random by the sport-fishing group.

A. Describe the distribution of F.

b. Fish, such as brown trout, become stressed once water temperatures exceed 18.3°C. Determine the probability Fwill exceed 18.3°C. Show your work.

A large metropolitan water district that uses water diverted from the same river is also planning to measure the water temperature at 100 randomly selected locations along the river. Because of sampling variability, the sample collected by the water district is likely to yield a sample mean water temperature different from the sample mean water temperature collected by the sport-fishing group. Let the random variable D represent the mean water temperature, in degrees Celsius, of 100 locations along the river selected at random by the large metropolitan water district.
c. Describe the distribution of D - F.

Answer 1

The distribution of F is normally distributed with a mean of 18°C and a standard error of 0.25°C. The probability that F will exceed 18.3°C is approximately 11.51%. The distribution of D - F is normally distributed, with the standard error calculated from the variances of the two independent sample sizes.

Step-by-step explanation:

A. Describe the distribution of F

The distribution of F, which represents the mean water temperature of 64 randomly selected locations along the river, would follow a normal distribution according to the Central Limit Theorem. Since the overall mean water temperature is 18°C and the standard deviation is 2°C, for a sample size of 64 the standard error would be the standard deviation divided by the square root of the sample size (2/√64 = 0.25). Therefore, the distribution of F would be normally distributed with a mean (μ) of 18°C and a standard error (σ/√64) of 0.25°C.

B. Probability F will exceed 18.3°C

To determine this probability, we first standardize the cutoff temperature using the Z-score formula: Z = (X - μ)/(σ/√64) = (18.3 - 18)/0.25 = 1.2. We look up the Z-score in a standard normal distribution table or use a calculator, which gives us a probability of approximately 0.1151, or 11.51%. Thus, there is an 11.51% chance that the mean water temperature at the 64 randomly selected locations will exceed 18.3°C.

C. Describe the distribution of D - F

The distribution of D - F, which represents the difference between the mean water temperatures collected by the water district and the sport-fishing group, would also be normally distributed. The standard error of the difference would be calculated by taking the square root of the sum of the variances of the two samples, considering that each sample has a different size, and therefore a different variance. Thus, the standard error for D would be 2/√100 = 0.2°C, and for F it is 0.25°C. So, the standard error for D - F would be √(0.2^2 + 0.25^2).

Answer 2

The distribution of F, the mean water temperature, is described as normal. The probability that F will exceed 18.3°C is approximately 0.9641. The distribution of D - F, the difference between the mean water temperatures measured by different groups, is also normal.

Step-by-step explanation:

A. The distribution of F, the mean water temperature at 64 randomly selected locations along the river, can be described as a normal distribution. Since the water temperature at each location is normally distributed with a mean of 18°C and a standard deviation of 2°C, the distribution of the sample mean temperature (F) will also be normally distributed.

B. To determine the probability that F will exceed 18.3°C, we can use the z-score formula. The z-score is calculated as (18.3 - 18) / (2 / sqrt(64)), which equals 1.8. Using a standard normal distribution table, we can find the probability associated with a z-score of 1.8, which is approximately 0.9641.

C. The distribution of D - F, the difference between the mean water temperature measured by the sport-fishing group (F) and the large metropolitan water district (D), can be described as a normal distribution. Since both F and D are normally distributed and independent, the difference between them will also be normally distributed.