Confidence Interval for a Mean

Confidence Interval for a Mean

When the Population Standard Deviation $\bco\sigma$ is KNOWN

When the Population Standard Deviation $\bco\sigma$ is UNKNOWN

• $s$ is the sample standard deviation
• $t^\star_{n-1}$ is the multiplier or critical value from the t-distribution with $n-1$ degrees of freedom

What is the difference between a z-distribution and a t-distribution?

Finding the critical value, $\bco{t_{n-1}^{\ast}}$ :

• Step 1: the degree of freedom is: $n-1=19-1=18$$\rightarrow$ found on the left column of the t-table
• Step 2: locate the confidence level $\colorFour C$ at the bottom row of the t-table, in particular 95%.
• Step 3: determine the critical value, $t^\star_{18}=2.101$ .

• Population:

  All households in Yaletown

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• Sample:

  30 households in Yaletown

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• Parameter:

  Average utility bill per month for all households in Yaletown

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• Statistic:

  $140, the average utility bill per month based on the sample of 30 households in Yaletown

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Interpretation of a confidence interval

• CORRECT: “We are 90% confident that the true average utility bill for all households in Yaletown is between $117 and $163 per month.”
• WRONG: “About 90% of households in Yaletown had utility bills between $117 and $163 per month.”
• WRONG: “There is a 90% probability that the true average utility bill is between $117 and $163 per month per month.”

Example: Confidence Interval for a Mean