Questions: Sound it out: Phonics is an instructional method in which children are taught to connect sounds with letters or groups of letters. A sample of 133 first-graders who were learning English were asked to identify as many letter sounds as possible in a period of one minute. The average number of letter sounds identified was 34.21 with a standard deviation of 23.33 Part 1 of 2 (a) Construct a 99.9% confidence interval for the mean number of letter sounds identified in one minute. Round the answers to at least two decimal places

Sound it out: Phonics is an instructional method in which children are taught to connect sounds with letters or groups of letters. A sample of 133 first-graders who were learning English were asked to identify as many letter sounds as possible in a period of one minute. The average number of letter sounds identified was 34.21 with a standard deviation of 23.33

Part 1 of 2
(a) Construct a 99.9% confidence interval for the mean number of letter sounds identified in one minute. Round the answers to at least two decimal places

Transcript text: Sound it out: Phonics is an instructional method in which children are taught to connect sounds with letters or groups of letters. A sample of 133 first-graders who were learning English were asked to Identify as many letter sounds as possible in a period of one minute. The average number of letter sounds identified was 34.21 with a standard deviation of 23.33 Part 1 of 2 (a) Construct a $99.9 \%$ confidence interval for the mean number of letter sounds identified in one minute. Round the answers to at least two decimal places

Solution

Solution Steps

Step 1: Given Data

We have a sample of $ n = 133 $ first-graders who identified an average of $ \bar{x} = 34.21 $ letter sounds in one minute, with a standard deviation of $ s = 23.33 $.

Step 2: Determine the Confidence Level

We are tasked with constructing a $ 99.9\% $ confidence interval for the mean number of letter sounds identified. The significance level $ \alpha $ is calculated as: \[ \alpha = 1 - 0.999 = 0.001 \]

Step 3: Calculate the Margin of Error

To find the margin of error, we use the formula: \[ E = z \cdot \frac{s}{\sqrt{n}} \] where $ z $ is the z-score corresponding to the $ 99.9\% $ confidence level. For $ 99.9\% $, $ z \approx 3.29 $.

Substituting the values: \[ E = 3.29 \cdot \frac{23.33}{\sqrt{133}} \approx 3.29 \cdot 2.0201 \approx 6.64 \]

Step 4: Construct the Confidence Interval

The confidence interval is given by: \[ \bar{x} \pm E \] Calculating the lower and upper bounds: \[ \text{Lower bound} = 34.21 - 6.64 \approx 27.55 \] \[ \text{Upper bound} = 34.21 + 6.64 \approx 40.87 \]