Question

Emissions of sulfur dioxide by industry set off chemical changes in the atmosphere that result in "acid rain." The acidity of liquids is measured by pH on a scale of 0 to 14. Distilled water has pH 7.0, and lower pH values indicate acidity. Normal rain is somewhat acidic, so acid rain is sometimes defined as rainfall with a pH below 5.0. A sample of 105 rainwater specimens had mean pH 5.43; standard deviation 0.54; and five‑number summary 4.33, 5.05, 5.44, 5.79, 6.81.

(1) Compare the mean and median and also the distances of the two quartiles from the median.

Does it appear that the distribution is quite symmetric? Why?

a) The third quartile is closer to the median than the first quartile is. This indicates that the third quartile may have been pushed down by a right‑hand skew in the distribution.

b)The median is a relatively resistant measure, while the mean isn't. Therefore, the mean gets pulled further away from the direction of a long tail. Since the mean is lower than the median, we conclude that the distribution has a right‑hand tail.

c) The mean is almost identical to the median, and the quartiles are similar distances from the median. This suggests that the distribution is reasonably symmetric.

d) The mean is a relatively resistant measure, while the median isn't. Therefore, the median gets pulled further away from the direction of a long tail. Since the median is lower than the mean, we conclude that the distribution has a right hand tail.

(2a) If the distribution is really ?(5.43,0.54), what proportion of observations would be less than 5.05? (Enter your answer rounded to four decimal places.)

(2b)If the distribution is really ?(5.43,0.54), what proportion of observations would be less than 5.79? (Enter your answer rounded to four decimal places.)  

(3) Do these proportions suggest that the distribution is close to Normal? Why?

a) These proportions do not provide information about Normality.

b)No, because the Normal distribution has a lower first quartile and a lower third quartile.

c)Yes, because the quartiles are very close to the quartiles of the Normal distribution.

d)No, because the Normal distribution has a higher third quartile and a higher first quartile.

Answer 1

mean pH =5.43

standard deviation =0.54

Five‑number summary is:

Minimum =4.33,

First quartile, Q1 =5.05,

Median =5.44,

Third quartile, Q3 =5.79,

Maximum =6.81

(1)

Mean =5.43 and Median =5.44. So, the mean and median are very close to each other.

Q3 - Median =5.79 - 5.44 =0.35

Median - Q1 =5.44 - 5.05 =0.39

So, the distances of both the quartiles from the median is similar.

Thus, option c) is correct. "The mean is almost identical to the median, and the quartiles are similar distances from the median. This suggests that the distribution is reasonably symmetric".

(2a)

X =5.05; =5.43; s =0.54

Z =(X - ​​​​​​)/s =(5.05 - 5.43)/0.54 = -0.7037

P(X < 5.05) =P(Z < -0.7037) =0.2408

(2b)

X =5.79

Z =(5.79 - 5.43)/0.54 =0.6667

P(X < 5.79) =P(Z < 0.6667) =0.7475

(3)

0.2408 =24.08% is close to 25% and 25% of the data falls below Q1 (in case of (2a) above, X =5.05 is Q1).

0.7475 =74.75% is close to 75% and 75% of the data falls below Q3 (in case (2b) above, X =5.79 is Q3).

Thus, option c) is correct. "Yes, because the quartiles are very close to the quartiles of the Normal distribution".