Question

In: Statistics and Probability

You measured dissolved argon concentrations in 20 water streams located at a region. The data is...

You measured dissolved argon concentrations in 20 water streams located at a region. The data is tabulated below. Assume normal distribution.

Observation no

Concentration

mg/L

Observation no

Concentration

mg/L

1

3.5

11

3.1

2

2.7

12

6.7

3

5.3

13

2.3

4

1.4

14

4.6

5

1.7

15

2.5

6

0.9

16

2.3

7

2.8

17

4.1

8

1.6

18

2.6

9

8.7

19

5.3

10

4.2

20

8.6

(a)  Find a 95% confidence interval on the mean dissolved argon concentration.

(b) Find a 95% prediction interval on the dissolved argon concentration for the next stream in the system that will be tested.

(c) Find an interval that will contain 95% of the values of the dissolved argon concenration with 95% confidence.

(d) Explain the difference in three intervals computed in parts (a), (b) and (c).

You are interested in knowing whether these data support a claim that mean dissolved argon concentration is 2 mg/L.

(e) Formulate an appropriate hypothesis testing procedure to investigate this claim.

(f) Test this hypotheses and draw conclusions, using α=0.02. Use P-value for this test.

Solutions

Expert Solution

Answers-

(a)  Find a 95% confidence interval on the mean dissolved argon concentration.

Let the 2 limits of the confidence interval be c1 and c2 and the interval [c1,c2] within which the unknown value of the population parameter is expected to lie is called the confidence interval and 95% which is also equal to 1- is also called the confidence coefficient.

Then, z= ( - ) / / Follows N(0,1)

P(-1.96 z 1.96) = 0.95

P(-1.96 ( - ) / / 1.96) =0.95

P( -1.96/   +1.96/) = 0.95

Thus,  1.96/ are the 95% confidence limits for the Mean parameter and the interval ( -1.96/ , +1.96/) is called the 95% confidence interval.

Substituting Values from the attached picture, the 95% confirnce interval is ( 3.75 -1.96 * 2.18/, 3.75 +1.96 * 2.18/) = (2.79, 4.70)

b) Find a 95% prediction interval on the dissolved argon concentration for the next stream in the system that will be tested.

In the same way as above, A prediction interval [l,u] for a future observation X in a normal distribution N(µ,σ2) is written as P[l < X< u]

This can be transformed to Z by subtracting with Mean and dividing with standard deviation.

P [(l - ) / < (x - ) / < (u - ) / ]

Hence,

(l - ) / = -z and (u - ) / =z

l = -z and u = + z

Thus, the prediction interval is written as ( -z ,   + z)

at 95% confidence, z= 1.96

Substituting the values ( 3.75 - 1.96*2.18 , 3.75 + 1.96*2.18) = (-0.53, 8.02)

Prediction interval will give the interval in which the next observation will fall.

c) Find an interval that will contain 95% of the values of the dissolved argon concenration with 95% confidence.

The interval( -1.96,1.96) is the interval since 95% of the area under the curve falls within this interval.

For a 95% confidence interval covers 95% of the normal curve. The probability of a value falling in the rejection region is less than 0.05. Since it is a symmetric curve, the probability of the value falling on the 2 sides of the rejection region is 0.05/2 = 0.025.

If p = 0.025, the value z* such that P(Z > z*) = 0.025, or P(Z < z*) = 0.975, is equal to 1.96. For a confidence interval with level C = 0.95, the value p is equal to (1-0.95)/2. A 95% confidence interval for the standard normal distribution, then, is the interval (-1.96, 1.96), since 95% of the area under the curve falls within this interval.

d) Explanation already covered in the first 3 parts a, b and c.

e) Testing of Hypothesis

Null Hypothesis H0: The mean dissolved Argan is 2mg/l

Alternative Hypothesis H1: The mean dissolved Argan is not 2mg/L

f) At 0.02 Level of significance.

Z = ( (Mean - ) / )

Cal Z = (3.75 - 2)/0.49

Cal Z = 3.57

Tab Z at 0.02 is 2.326

Thus, Cal Z > Tab Z, this gives us significant evidence to reject the Null Hypothesis. This means that the data does not support the claim that the mean value of the Argan is 2mg/L


Related Solutions

The quantity of dissolved oxygen is a measure of water pollution in lakes, rivers, and streams....
The quantity of dissolved oxygen is a measure of water pollution in lakes, rivers, and streams. Water samples were taken at four different locations in a river in an effort to determine if water pollution varied from location to location. Location I was 500 meters above an industrial plant water discharge point and near the shore. Location II was 200 meters above the discharge point and in midstream. Location III was 50 meters downstream from the discharge point and near...
The quantity of dissolved oxygen is a measure of water pollution in lakes, rivers, and streams....
The quantity of dissolved oxygen is a measure of water pollution in lakes, rivers, and streams. Water samples were taken at four different locations in a river in an effort to determine if water pollution varied from location to location. Location I was 500 meters above an industrial plant water discharge point and near the shore. Location II was 200 meters above the discharge point and in midstream. Location III was 50 meters downstream from the discharge point and near...
The osmotic pressure of an unknown substance dissolved in 100 mL of water is measured at...
The osmotic pressure of an unknown substance dissolved in 100 mL of water is measured at 298 K. The concentration of this substance is 25.5 kg m–3 and the osmotic pressure is 4.50 x 104 Pa. a) Determine the molecular weight of this substance. b) Calculate the change in the freezing point of water caused by the addition of this substance. (Keep in mind that the density of a dilute aqueous solution is approximately 1 g/mL).
1. 20 g of MgCO3 are dissolved in water and brought to a final volume of...
1. 20 g of MgCO3 are dissolved in water and brought to a final volume of 700 mL. What is the molar concentration of the solution? 2. Calculate the normality of the solution. 3. If the MgCO3 completely dissociates to Mg2+ and CO3 2- , what are the concentrations of Mg2+ and CO3 2- (in g/L) in the solution? 4. Calculate the mass fraction of Mg2+ in the solution. 5. Calculate the molar fraction of C in solution.
According to test data, a new type of engine takes in streams of water at 200...
According to test data, a new type of engine takes in streams of water at 200 °C, 3 bar and 100 °C, 3 bar. The inlet ratio of the mass flow rate of the higher temperature stream to lower temperature stream is 4. A single stream exits at 3.0 bar with a mass flow rate of 5400 kg/h. There is no significant heat transfer between the engine and its surroundings, and kinetic and potential energy effects are negligible. For operation...
M14 #20: The quantity of dissolved oxygen is a measure of water pollution in lakes, rivers,...
M14 #20: The quantity of dissolved oxygen is a measure of water pollution in lakes, rivers, and streams. Water samples were taken at four different locations in a river in an effort to determine if water pollution varied from location to location. Location I was 500 meters above an industrial plant water discharge point and near the shore. Location II was 200 meters above the discharge point and in midstream. Location III was 50 meters downstream from the discharge point...
DSW is a midsized coal mining company with 20 mines located in Hessen region in central...
DSW is a midsized coal mining company with 20 mines located in Hessen region in central Germany. The company operates deep mines as well as strip mines. Most of the coal mined is sold under contract, with excess production sold on the spot market. The coal mining industry, especially high-sulfur coal operations such as DSW, has been hard-hit by environmental regulations and warmer than expected winter 2014/2015 and 2016/2017. Recently, however, a combination of increased demand for coal and new...
Provided set of data are concentrations of suspended solids in river water that are collected for...
Provided set of data are concentrations of suspended solids in river water that are collected for environmental characteristics. Total of 36 samples have concentrations of: 84 49 61 40 83 67 45 66 70 69 80 58 68 60 67 72 73 70 57 63 70 78 52 67 53 67 75 61 70 81 76 79 75 76 58 31 a) Construct a stem‐and‐leaf display of the data. b) Calculate first and third quartiles and median from stem‐and‐leaf display...
A 200 gallon tank initial contains 100 gallons of water and 20 pounds of dissolved salt....
A 200 gallon tank initial contains 100 gallons of water and 20 pounds of dissolved salt. Brine solution begins to enter the tank at the rate of 2 gal/min with a salt concentration of 2 lb/gal. The well mixed solution leaves the tank at the rate of 1 gal/min. Find the amount of salt inside the tank 50 minutes after the process starts?
You ground your vitamin C tablet and dissolved it in 250. mL of water in a...
You ground your vitamin C tablet and dissolved it in 250. mL of water in a volumetric flask. You used a 25.0 mL aliquot of this solution for your titration. If it takes 11.50 milliliters of your 0.0151M titrant, how many milligrams of ascorbic acid (MW 176.12) were in the original tablet?
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT