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Answer :
To solve this problem, we're going to calculate four specific quantities related to the large-counts condition from two independent samples. The large-counts condition requires these calculations to ensure that each count is at least 10, which supports the assumptions for approximate normality in statistical procedures.
Let's break it down step-by-step:
1. Given Values:
- Sample size for the first group, [tex]\( n_1 = 325 \)[/tex].
- Sample size for the second group, [tex]\( n_2 = 455 \)[/tex].
- Proportion estimate for the first group, [tex]\( \hat{\beta}_1 = 0.71 \)[/tex].
- Proportion estimate for the second group, [tex]\( \hat{\beta}_2 = 0.64 \)[/tex].
2. Required Quantities:
- Compute [tex]\( n_1 \times \hat{\beta}_2 \)[/tex].
- Compute [tex]\( n_1 \times (1 - \hat{\beta}_1) \)[/tex].
- Compute [tex]\( n_2 \times \hat{\beta}_2 \)[/tex].
- Compute [tex]\( n_2 \times (1 - \hat{\beta}_2) \)[/tex].
Now, let's do each calculation:
- First Calculation: [tex]\( n_1 \times \hat{\beta}_2 \)[/tex]
- Substitute [tex]\( n_1 = 325 \)[/tex] and [tex]\( \hat{\beta}_2 = 0.64 \)[/tex].
- Calculation: [tex]\( 325 \times 0.64 = 208.0 \)[/tex].
- Second Calculation: [tex]\( n_1 \times (1 - \hat{\beta}_1) \)[/tex]
- Substitute [tex]\( n_1 = 325 \)[/tex] and [tex]\( \hat{\beta}_1 = 0.71 \)[/tex].
- Calculation: [tex]\( 325 \times (1 - 0.71) = 325 \times 0.29 = 94.25 \)[/tex].
- Third Calculation: [tex]\( n_2 \times \hat{\beta}_2 \)[/tex]
- Substitute [tex]\( n_2 = 455 \)[/tex] and [tex]\( \hat{\beta}_2 = 0.64 \)[/tex].
- Calculation: [tex]\( 455 \times 0.64 = 291.2 \)[/tex].
- Fourth Calculation: [tex]\( n_2 \times (1 - \hat{\beta}_2) \)[/tex]
- Substitute [tex]\( n_2 = 455 \)[/tex] and [tex]\( \hat{\beta}_2 = 0.64 \)[/tex].
- Calculation: [tex]\( 455 \times (1 - 0.64) = 455 \times 0.36 = 163.8 \)[/tex].
These computed values are:
- [tex]\( n_1 \times \hat{\beta}_2 = 208.0 \)[/tex]
- [tex]\( n_1 \times (1 - \hat{\beta}_1) = 94.25 \)[/tex]
- [tex]\( n_2 \times \hat{\beta}_2 = 291.2 \)[/tex]
- [tex]\( n_2 \times (1 - \hat{\beta}_2) = 163.8 \)[/tex]
Each of these values is indeed greater than 10, satisfying the large-counts condition. This ensures that the sample sizes are sufficient for the statistical procedures that assume normal approximation.
Let's break it down step-by-step:
1. Given Values:
- Sample size for the first group, [tex]\( n_1 = 325 \)[/tex].
- Sample size for the second group, [tex]\( n_2 = 455 \)[/tex].
- Proportion estimate for the first group, [tex]\( \hat{\beta}_1 = 0.71 \)[/tex].
- Proportion estimate for the second group, [tex]\( \hat{\beta}_2 = 0.64 \)[/tex].
2. Required Quantities:
- Compute [tex]\( n_1 \times \hat{\beta}_2 \)[/tex].
- Compute [tex]\( n_1 \times (1 - \hat{\beta}_1) \)[/tex].
- Compute [tex]\( n_2 \times \hat{\beta}_2 \)[/tex].
- Compute [tex]\( n_2 \times (1 - \hat{\beta}_2) \)[/tex].
Now, let's do each calculation:
- First Calculation: [tex]\( n_1 \times \hat{\beta}_2 \)[/tex]
- Substitute [tex]\( n_1 = 325 \)[/tex] and [tex]\( \hat{\beta}_2 = 0.64 \)[/tex].
- Calculation: [tex]\( 325 \times 0.64 = 208.0 \)[/tex].
- Second Calculation: [tex]\( n_1 \times (1 - \hat{\beta}_1) \)[/tex]
- Substitute [tex]\( n_1 = 325 \)[/tex] and [tex]\( \hat{\beta}_1 = 0.71 \)[/tex].
- Calculation: [tex]\( 325 \times (1 - 0.71) = 325 \times 0.29 = 94.25 \)[/tex].
- Third Calculation: [tex]\( n_2 \times \hat{\beta}_2 \)[/tex]
- Substitute [tex]\( n_2 = 455 \)[/tex] and [tex]\( \hat{\beta}_2 = 0.64 \)[/tex].
- Calculation: [tex]\( 455 \times 0.64 = 291.2 \)[/tex].
- Fourth Calculation: [tex]\( n_2 \times (1 - \hat{\beta}_2) \)[/tex]
- Substitute [tex]\( n_2 = 455 \)[/tex] and [tex]\( \hat{\beta}_2 = 0.64 \)[/tex].
- Calculation: [tex]\( 455 \times (1 - 0.64) = 455 \times 0.36 = 163.8 \)[/tex].
These computed values are:
- [tex]\( n_1 \times \hat{\beta}_2 = 208.0 \)[/tex]
- [tex]\( n_1 \times (1 - \hat{\beta}_1) = 94.25 \)[/tex]
- [tex]\( n_2 \times \hat{\beta}_2 = 291.2 \)[/tex]
- [tex]\( n_2 \times (1 - \hat{\beta}_2) = 163.8 \)[/tex]
Each of these values is indeed greater than 10, satisfying the large-counts condition. This ensures that the sample sizes are sufficient for the statistical procedures that assume normal approximation.
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