To determine the expected frequencies for a sample size of 1,568 based on the provided percent distribution of the US population by age group, we follow these steps:
1. Identify the Sample Size and Percent Distribution:
- Sample Size ([tex]\( n \)[/tex]) = 1,568
- Percent distribution:
- Over 65 years: [tex]\( 12.43\% \)[/tex]
- 45-65 years: [tex]\( 22.01\% \)[/tex]
- 25-44 years: [tex]\( 30.22\% \)[/tex]
- 15-24 years: [tex]\( 13.93\% \)[/tex]
- Under 15 years: [tex]\( 21.41\% \)[/tex]
2. Convert Percentages to Proportions:
- Convert each percentage into a proportion by dividing by 100.
For "Over 65 years":
[tex]\[
12.43\% = 0.1243
\][/tex]
For "45-65 years":
[tex]\[
22.01\% = 0.2201
\][/tex]
For "25-44 years":
[tex]\[
30.22\% = 0.3022
\][/tex]
For "15-24 years":
[tex]\[
13.93\% = 0.1393
\][/tex]
For "Under 15 years":
[tex]\[
21.41\% = 0.2141
\][/tex]
3. Calculate the Expected Frequencies:
- Multiply each proportion by the sample size ([tex]\( n = 1,568 \)[/tex]) to obtain the expected frequency for each age group:
For "Over 65 years":
[tex]\[
0.1243 \times 1568 = 194.9024
\][/tex]
For "45-65 years":
[tex]\[
0.2201 \times 1568 = 345.1168
\][/tex]
For "25-44 years":
[tex]\[
0.3022 \times 1568 = 473.8496
\][/tex]
For "15-24 years":
[tex]\[
0.1393 \times 1568 = 218.4224
\][/tex]
For "Under 15 years":
[tex]\[
0.2141 \times 1568 = 335.7088
\][/tex]
4. Present the Expected Frequencies:
We can now fill in the expected frequencies in the table:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
\text{Age Group} & \text{Expected Frequency} \\
\hline
\text{Over 65 years} & 194.9024 \\
\hline
\text{45-65 years} & 345.1168 \\
\hline
\text{25-44 years} & 473.8496 \\
\hline
\text{15-24 years} & 218.4224 \\
\hline
\text{Under 15 years} & 335.7088 \\
\hline
\end{array}
\][/tex]
5. Formulating the Null Hypothesis:
The most appropriate null hypothesis (H₀) based on the data comparison would be:
[tex]\[
\text{The distribution of ages among people who are left-handed is the same as that provided by the census data.}
\][/tex]
