A research firm conducted a survey to determine the mean...

A research firm conducted a survey to determine the mean amount steady smokers spend on cigarettes during a week. A sample of 49 steady smokers revealed that $\bar{X}=\$ 20$ and $s=\$ 5$ a. What is the point estimate of the population mean? Explain what it indicates. b. Using the 95 percent level of confidence, determine the confidence interval for $\mu .$ Explain what it indicates.

A research firm conducted a survey to determine the mean amount steady smokers spend on cigarettes during a week. A sample of 49 steady smokers revealed that $\bar{X}=\$ 20$ and $s=\$ 5$
a. What is the point estimate of the population mean? Explain what it indicates.
b. Using the 95 percent level of confidence, determine the confidence interval for $\mu .$ Explain what it indicates.

Key Concepts

Point Estimate

A point estimate is a single value, derived from sample data, that is used to approximate an unknown population parameter. In statistics, the sample mean is a common point estimate of the population mean, offering the best single-value approximation based on the observed data.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter with a specified level of confidence. It provides not only an estimate of the parameter but also an indication of the uncertainty associated with the estimate.

Standard Error

The standard error measures the variability of a sample statistic, such as the sample mean, from sample to sample. It is computed as the sample standard deviation divided by the square root of the sample size, and it quantifies the precision of the point estimate.

Margin of Error

The margin of error represents the extent of the interval on either side of the point estimate, within which the true population parameter is expected to lie with a given level of confidence. It is determined by multiplying the critical value from the appropriate statistical distribution by the standard error.

Level of Confidence

The level of confidence expresses the probability that the confidence interval constructed from the sample data will contain the true population parameter. For instance, a 95 percent confidence level indicates that if the same sampling procedure were repeated numerous times, approximately 95 percent of the constructed intervals would include the true parameter.

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