Respuesta :
Answer:
a. The number of people that should be in the pilot study are 600 people
b. The point estimate is 0.62[tex]\overline 6[/tex]
c. At 95% confidence level the true population proportion of potential car buyers of hybrid vehicle is between the confidence interval (0.588, 0.6654)
d. Two ways to reduce the margin of error are;
1) Reduce the confidence interval
2) Use a larger sample size
Step-by-step explanation:
a. The given parameters for the estimation of sample size is given as follows;
The margin of error for the confidence interval, E = 4% = 0.04
The confidence level = 95%
The sample size formula for a proportion as obtained from an online source is given as follows;
[tex]n = \dfrac{Z^2 \times P \times (1 - P)}{E^2}[/tex]
Where, P is the estimated proportions of the desired statistic, therefore, we have for a new study, P = 0.5;
Z = The level of confidence at 95% = 1.96
n + The sample size
Therefore, we have;
[tex]n = \dfrac{1.96^2 \times 0.5 \times (1 - 0.5)}{0.04^2} = 600.25[/tex]
Therefore, the number of people that should be in the pilot study in order to meet this goal at 95% confidence level is n = 600 people
b. The point estimate for the population proportion is the sample proportion given as follows;
[tex]\hat p = \dfrac{x}{n}[/tex]
Where;
x = The number of the statistic in the sample
n = The sample size
From the question, we have;
The number of potential car buyers, n = 600
The number of respondent in the sample that indicated that they would consider purchasing a hybrid, x = 376
Therefore, the point estimate, for the proportion of potential car buyers that would consider buying a hybrid vehicle, [tex]\hat p[/tex] = 376/600 = 0.62[tex]\overline 6[/tex]
c. The confidence interval for a proportion is given as follows
[tex]CI=\hat{p}\pm z\times \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}[/tex]
Therefore, we get;
[tex]CI=0.62 \overline 6\pm 1.96\times \sqrt{\dfrac{\hat{0.62 \overline 6}\cdot (1-\hat{0.62 \overline 6})}{600}}[/tex]
C.I. ≈ 0.6267 ± 0.0387
The 95% confidence interval for the true population proportion of potential buyers of hybrid vehicle, C.I. = (0.588, 0.6654)
d. The margin of error is given by the following formula;
[tex]MOE_\gamma = z_\gamma \times \sqrt{\dfrac{\sigma ^2}{n} }[/tex]
Where;
[tex]MOE_\gamma[/tex] = Margin of error at a given level of confidence
[tex]z_\gamma[/tex] = z-score
σ = The standard deviation
n = The sample size
Therefore, the margin error can be reduced by the following two ways;
1) Reducing the confidence interval and therefore, the z-score
2) Increasing the sample size
