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 00 to 14.14. Distilled water has pH 7.0,7.0, and lower pH values indicate acidity. Normal rain is somewhat acidic, so acid rain is sometimes defined as rainfall with a pH below 5.0.5.0.

A sample of 105105 rainwater specimens had mean pH 5.43;5.43; standard deviation 0.54;0.54; and five‑number summary 4.33,4.33, 5.05,5.05, 5.44,5.44, 5.79,5.79, 6.81.6.81.

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

b. 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.

c. 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.

d. 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.

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

proportion:

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

proportion:

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

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

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

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

d. These proportions do not provide information about Normality.

Answer 1

Given: Five-number summary

Minimum = 4.33 1st Quartile   = 5.05 Median = 5.44 3rd Quartile = 5.79 Maximum = 6.81

Mean = 5.43 Standard Deviation = 0.54

For symmertic data, usually, the mean and median are approximately equal. Also, the median would be almost equally distanced from the 1st and 3rd quartiles.

====================================================================

a) Comparing the mean and median: We find that the mean (=5.43) and Median (=5.44) are approximately equal.  Also the distances of the median from the two quartiles Q1 and Q3 are obtained are (5.44 - 5.05 ) = 0.39 and (5.79 - 5.44 ) = 0.35 respectively, which when rounded to one decimal, (= 0.4) are equal. The correct option would be:

a. 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.

===================================================================

(b)  If the distribution is really ?(5.43,0.54), the proportion of observations less than 5.05 = ?

Since,

From standard normal table,

= 0.24196

Hence, the proportion of observations less than 5.05 = 0.2420

===================================================================

If the distribution is really ?(5.43,0.54), the proportion of observations less than 5.79 = ?

Since,

= 0.74857

Hence, the proportion of observations less than 5.79 = 0.7486

====================================================================

We know that, for a Normal distribution, the proportion of observations less than the 1st Quartile must be 25% ( one fourth of the observations lie below Q1 ) and for the 3rd Quartile must be 75% ( one fourth of the observations lie below Q1 ). From the proportions obtained, i.e. 0.2420 and 0.7486, respectively,  we may infer that:

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