In a simple random sample of 8 o teenagers, the average...

In a simple random sample of 8 o teenagers, the average number of texts handled in a day was 50 with a standard deviation of $15 .$ What is the $96 \%$ confidence interval for the average number of texts handled by all teens daily? (A) $50 \pm 2.054(15)$ (B) $50 \pm 2.054 \frac{15}{\sqrt{79}}$ (C) $50 \pm 2.054 \frac{15}{\sqrt{80}}$ (D) $50 \pm 2.054 \frac{15}{\sqrt{80}}$ (E) $50 \pm 2.054 \frac{15}{\sqrt{80}}$

In a simple random sample of 8 o teenagers, the average number of texts handled in a day was 50 with a standard deviation of $15 .$ What is the $96 \%$ confidence interval for the average number of texts handled by all teens daily?
(A) $50 \pm 2.054(15)$
(B) $50 \pm 2.054 \frac{15}{\sqrt{79}}$
(C) $50 \pm 2.054 \frac{15}{\sqrt{80}}$
(D) $50 \pm 2.054 \frac{15}{\sqrt{80}}$
(E) $50 \pm 2.054 \frac{15}{\sqrt{80}}$

Key Concepts

Margin of Error

The margin of error is the amount added to and subtracted from the sample mean to create the confidence interval. It is obtained by multiplying the critical value (from the t-distribution when the population standard deviation is unknown) by the standard error, representing the maximum expected difference between the sample statistic and the true population parameter.

Confidence Interval

A confidence interval is a range of values derived from sample data that is likely to contain the true population parameter with a specified level of confidence. It provides an estimate of the uncertainty associated with a sample statistic, such as the mean, and indicates how precisely this statistic estimates the parameter of interest.

t?Distribution

The t-distribution is used when estimating the population mean from a small sample size and the population standard deviation is unknown. It accounts for additional variability due to using the sample standard deviation as an estimate and is characterized by degrees of freedom, which typically equals the sample size minus one.

Degrees of Freedom

Degrees of freedom refer to the number of independent values that can vary in an analysis without breaking any constraints. In the context of the t-distribution for estimating a population mean, the degrees of freedom are calculated as the sample size minus one, and they affect the shape of the t-distribution and the corresponding critical values.

Standard Error

The standard error is a measure of the variability of the sample mean and is calculated by dividing the sample standard deviation by the square root of the sample size. It quantifies how much the sample mean is expected to vary from one sample to another, and it is a key component in constructing confidence intervals.

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