Property taxes are based on assessed values of property. In most states, the law requires that assessed values be “at or near” market value of the property.

In one Washington county, a tax protest group has claimed that assessed values are higher than market values. To address this claim, the county tax assessor, together with representatives from the protest group, has selected 15 properties at random that have sold within the past six months. Both parties agree that the sales price was the market value at the time of the sale.

The assessor then listed the assessed values and the sales values side by side, as shown.

House Assessed Value ($) Market Value ($)
1 302,000 198,000 2 176,000 182,400 3 149,000 154,300 4 198,500 198,500 5 214,000 218,000 6 235,000 230,000 7 305,000 298,900 8 187,500 190,000 9 150,000 149,800 10 223,000 222,000 11 178,500 180,000 12 245,000 250,900 13 167,000 165,200 14 219,000 220,700 15 334,000 320,000

a. Assuming that the population of assessed values and the population of market values have the same distribution shape and that they may differ only with respect to medians, state the appropriate null and alternative hypotheses.

b. Test the hypotheses using an a  0.01 level.

c. Discuss why one would not assume that the samples were obtained from normal distributions for this problem. What characteristic about the market values of houses would lead you to conclude that these data were not normally distributed?