Political party affiliation is believed to be a very strong indicator of how voters will vote in Presidential Elections. You are interested in determining if voter party loyalty has changed since 1992. During the 1992 election, the proportion of self-proclaimed Republicans who voted for George H. W. Bush was 0.91. During the 2012 election, in a survey of 257 Republican voters, 205 indicated that they had voted for Mitt Romney. The 90% confidence interval for this proportion is ( 0.7564 , 0.8389 ). What is the best conclusion you can make from this information that is listed below?

1) The proportion of self-proclaimed Republicans who voted for their party candidate in 2012 who changed from 0.91 is 90%.

2) We can conclude that the proportion of self-proclaimed Republicans who voted in the 2012 election for their party candidate is larger than 0.91.

3) We can claim that the proportion of self-proclaimed Republicans who voted in the 2012 election for their party candidate is smaller than 0.91.

4) The confidence interval does not provide enough information to form a conclusion.

5) We can not claim that the proportion of self-proclaimed Republicans who voted in the 2012 election for their party candidate differs from 0.91.

Correct option: (3).

Step-by-step explanation:

The hypothesis to determine whether the voter party loyalty has changed since 1992 is:

H₀: The voter party loyalty has not changed since 1992, i.e. p = 0.91.

Hₐ: The voter party loyalty has changed since 1992, i.e. p ≠ 0.91.

The decision rule is:

If the 90% confidence interval consists of the null hypothesis value, 0.91 then the null hypothesis will be accepted or else it will be rejected.

The 90% confidence interval for the proportion of self-proclaimed Republicans is:

(0.7564, 0.8389)

As the null hypothesis value, 0.91 is not included in the interval, it implies that the null hypothesis is rejected.

Conclusion:

As the null hypothesis was rejected at 10% level of significance it can be concluded that the voter party loyalty has changed since 1992.

Now consider the confidence interval (0.7564 , 0.8389).

The interval included values less than 0.91.

This implies that the proportion of self-proclaimed Republicans who voted in the 2012 election for their party candidate is smaller than 0.91.

Thus, the correct option is (3).

the mistake in the pattern is that :

b. the decimal point was moved one too many place values each time

the correct pattern could be made by :

b. moving the decimal point to the right the same numbers of place values as the exponents