Questions: A soft drink manufacturer wishes to know how many soft drinks adults drink each week. They want to construct a 95% confidence interval for the mean and are assuming that the population standard deviation for the number of soft drinks consumed each week is 1.1. The study found that for a sample of 949 adults the mean number of soft drinks consumed per week is 3.6. Construct the desired confidence interval. Round your answers to one decimal place. Lower endpoint: Upper endpoint

A soft drink manufacturer wishes to know how many soft drinks adults drink each week. They want to construct a 95% confidence interval for the mean and are assuming that the population standard deviation for the number of soft drinks consumed each week is 1.1. The study found that for a sample of 949 adults the mean number of soft drinks consumed per week is 3.6. Construct the desired confidence interval. Round your answers to one decimal place. Lower endpoint: Upper endpoint
Transcript text: A soft drink manufacturer wishes to know how many soft drinks adults drink each week. They want to construct a $95 \%$ confidence interval for the mean and are assuming that the population standard deviation for the number of soft drinks consumed each week is 1.1. The study found that for a sample of 949 adults the mean number of soft drinks consumed per week is 3.6. Construct the desired confidence interval. Round your answers to one decimat place. Answer How to enter your answer (opens in new window) Tables Keypad Keyboard Shortcuts Lower endpoint: $\square$ Upper endpoint $\square$
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Solution

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Solution Steps

Step 1: Calculate the Z-Score

For a \(95\%\) confidence level, the Z-score is given by: \[ Z = 1.96 \]

Step 2: Calculate the Margin of Error

The margin of error (\(E\)) is calculated using the formula: \[ E = Z \times \frac{\sigma}{\sqrt{n}} \] Substituting the known values: \[ E = 1.96 \times \frac{1.1}{\sqrt{949}} \approx 0.07 \] Thus, the margin of error is: \[ \text{Margin of Error} = 0.07 \]

Step 3: Construct the Confidence Interval

The confidence interval for the mean is given by: \[ \bar{x} \pm E \] Substituting the sample mean and the margin of error: \[ 3.6 \pm 0.07 \] This results in the confidence interval: \[ (3.6 - 0.07, 3.6 + 0.07) = (3.53, 3.67) \] Rounding to one decimal place, we have: \[ (3.5, 3.7) \]

Final Answer

The lower endpoint is \(3.5\) and the upper endpoint is \(3.7\). Thus, the final answer is: \[ \boxed{\text{Lower endpoint: } 3.5, \text{ Upper endpoint: } 3.7} \]

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