In a survey of 300 random households in a particular village, mother's have been asked if they used cellphone to communicate there are 76 who said YES, Use 90%, 95% and 99% confidence to estimate the proportion of all mothers who use the cellphone to communicate
1
Expert's answer
2022-05-02T16:33:28-0400
The sample proportion is computed as follows, based on the sample size N=300 and the number of favorable cases X=76:
p^=NX=30076≈0.253333
1. The critical value for α=0.10 is zc=z1−α/2=1.6449.
The corresponding confidence interval is computed as shown below:
CI(Proportion)=(p^−zcnp^(1−p^),
p^+zcnp^(1−p^))
=(30076−1.644930030076(1−30076),
30076+1.644930030076(1−30076))
=(0.212,0.295)
Therefore, based on the data provided, the 90% confidence interval for the population proportion is 0.212<p<0.295, which indicates that we are 90% confident that the true population proportion p is contained by the interval (0.212,0.295).
2. The critical value for α=0.05 is zc=z1−α/2=1.96.
The corresponding confidence interval is computed as shown below:
CI(Proportion)=(p^−zcnp^(1−p^),
p^+zcnp^(1−p^))
=(30076−1.9630030076(1−30076),
30076+1.9630030076(1−30076))
=(0.204,0.303)
Therefore, based on the data provided, the 95% confidence interval for the population proportion is 0.204<p<0.303, which indicates that we are 95% confident that the true population proportion p is contained by the interval (0.204,0.303).
3. The critical value for α=0.01 is zc=z1−α/2=2.5758.
The corresponding confidence interval is computed as shown below:
CI(Proportion)=(p^−zcnp^(1−p^),
p^+zcnp^(1−p^))
=(30076−2.575830030076(1−30076),
30076+2.575830030076(1−30076))
=(0.189,0.318)
Therefore, based on the data provided, the 99% confidence interval for the population proportion is 0.189<p<0.318, which indicates that we are 99% confident that the true population proportion p is contained by the interval (0.189,0.318).
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