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In a poll, [tex]$47 \%$[/tex] of members of political party [tex]$A$[/tex] and [tex]$40 \%$[/tex] of members of political party [tex]$B$[/tex] agreed with the statement "Obesity impacts individuals, but doesn't have a major impact on society." A [tex]$95 \%$[/tex] confidence interval for the difference in these proportions is [tex]$(-0.11$[/tex] to -0.03[tex]$)$[/tex], or in terms of percentage points, [tex]$(-11 \%$[/tex] to [tex]$-3 \%)$[/tex]. (Use political party B as population 1.)

Interpret this confidence interval. If the interval contains 0, indicate what this means. Explain the meaning of positive and/or negative values.

[tex]$\square$[/tex] The difference in [tex]$\square$[/tex] (party [tex]$B$[/tex] - party [tex]$A$[/tex]) is between [tex]$\square$[/tex] and [tex]$\square$[/tex] [tex]$\square$[/tex].
(Type integers or decimals. Do not round. Use ascending order.)

Sagot :

GinnyAnswer
Certainly! Let's interpret the given 95% confidence interval for the difference in proportions between the members of political party B and political party A.

Here are the steps:

1. Determine the Confidence Interval:
- We are given that the 95% confidence interval for the difference in proportions is [tex]\((-0.11\)[/tex] to [tex]\(-0.03)\)[/tex], or in terms of percentage points, [tex]\((-11\%\)[/tex] to [tex]\(-3\%\)[/tex]).

2. Check if the Interval Includes Zero:
- The interval ranges from [tex]\(-0.11\)[/tex] to [tex]\(-0.03\)[/tex]. Since 0 is not within this range (i.e., 0 is not between [tex]\(-0.11\)[/tex] and [tex]\(-0.03\)[/tex]), the confidence interval does not contain zero.

3. Interpret the Negative Values in the Interval:
- Because the confidence interval consists of only negative values, this indicates that, with 95% confidence, the proportion of members of political party B who agreed with the statement is lower than the proportion of members of political party A who agreed with the statement.

4. Summarize the Interpretation:
- The negative values suggest that party B has a lower agreement rate with the statement compared to party A.

Therefore, the interpretation of the confidence interval is as follows:

"The difference in proportions (party B - party A) is between [tex]\(-0.11\)[/tex] and [tex]\(-0.03\)[/tex]. Since the interval does not include 0, we can conclude at a 95% confidence level that the proportion of members of political party B who agree with the statement is lower than that of the members of political party A."

To fill in the blanks in the statement:
- "that the difference in proportions (party B - party A) is between [tex]\(\-0.11\)[/tex] and [tex]\(\-0.03\)[/tex][tex]\(\)[/tex]."

So, the completed statement is:
- “The difference in proportions (party B - party A) is between -0.11 and -0.03.”

Additionally,
- If the interval contains 0, it means that: the difference in proportions cannot be conclusively said to be different from zero at the 95% confidence level (i.e., there could be no significant difference).
- The meaning of positive and/or negative values: Negative values indicate that party B has a lower agreement rate than party A, while positive values would indicate the reverse.

Therefore, we can summarize that the interval never contains zero, indicating a clear difference in proportions, and in this case, party B’s agreement rate is lower than that of party A.
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