A research firmsurveyedy to determine the average amount smokers spend on cigarettes during a week. The company found that the distribution of amounts they spend per week tended to follow a normal distribution, with a standard deviation of $10. A sample of 49 smokers revealed that the mean weekly expenditure was $55. What is the point estimator of the population mean? Explain your answer. At the 95% confidence level, determine the confidence interval of the population mean. Explain its significance.
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The point estimator of the population mean is the sample mean. In this case, the sample mean is $55.
A 95% confidence interval for the population mean can be calculated using the formula:
Confidence Interval = (sample mean) ± (critical value) * (standard deviation / sqrt(sample size))
The critical value for a 95% confidence interval can be found using a t-table or a t-distribution calculator. Since the sample size is large (n = 49), a z-score can also be used instead of a t-score.
Using the z-table, the critical value for a 95% confidence interval is approximately 1.96.
Substituting the given values into the formula:
Confidence Interval = $55 ± 1.96 * ($10 / sqrt(49))
= $55 ± 1.96 * $1.42
≈ $55 ± $2.78
Therefore, the 95% confidence interval for the population mean is approximately $52.22 to $57.78.
This confidence interval indicates that we are 95% confident that the true population mean lies within the range of $52.22 to $57.78. In other words, if we were to repeat this study multiple times and calculate 95% confidence intervals each time, we would expect that about 95% of the intervals would contain the true population mean.
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