Question

Children Watching TV. The A. C. Nielsen Company reported in the Nielsen Report on Television that the mean weekly television viewing time for children aged $2-6$ years is $24.85$ hours. Assume that the weekly television viewing times of such children are generally distributed with a standard deviation of $6.23$ hours and apply the empirical rule to fill in the blanks.

a. Approximately $68\%$ of all such children watch between ---- and ----- hours of TV per week.

b. Approximately $95\%$ of all such children watch between ---- and ---- hours of TV per week.

c. Approximately $99.7\%$ of all such children watch between ---and ----- hours of TV per week.

d. Draw graphs similar to those in Fig. $6.27$on page $272$to portray your results.

Short answer

a). Approximately $68\%$of all such children watch between $18.62$and $31.08$hours of TV per week.

b). Approximately $95\%$of allsuch children watch between $\underset{¯}{12.39and}37.31$hours of TV per week.

c). Approximately $99.7\%$ of all such children watch between $\underset{¯}{6.16}$ and $\underset{¯}{43.54}$ hours of TV per week.

d). The graphs are shown below.

Step-by-step solution

Part (a) Step 1: Given Information

The given statement is "Approximately $68\%$ of all such children watch between ---- and ---- hours of TV per week."

The average weekly television watching time is $\mu =24.85$ hours, with a standard deviation of $\sigma =6.23hours$.

Part (a) Step 2: Explanation

By using proper empirical rule, approximately $68\%$of values lie in the interval $(\mu -\sigma ,\mu +\sigma )$.

role="math" localid="1653127139227" $(\mu -\sigma ,\mu +\sigma )$$=(24.85-6.23,24.85+6.23)$

$=(18.62,31.08)$

Approximately $68\%$ of all such children watch between $18.62$ and $31.08$ hours of TV per week.

Part (b) Step 1: Given Information

The average weekly television watching time is $\mu =24.85$ hours, with a standard deviation of $\sigma =6.23$ hours.

Part (b) Step 2: Explanation

Approximately $95\%$ of values fall within the interval, according to the empirical rule. $(\mu -2\sigma ,\mu +2\sigma )$.

$(\mu -2\sigma ,\mu +2\sigma )$$=(24.85-12.46,24.85+12.46)$

$=(12.39,37.31)$

Hence, approximately $95\%$of all such children watch between $12.39$and$37.31$hours of TV per week.

Part (c) Step 1: Given Information

The average weekly television watching time is $64.4$ hours, with a standard deviation of $24$ hours.

Part (c) Step 2: Explanation

According to the empirical rule, approximately $99.7\%$of values fall within the range. $(\mu -3\sigma ,\mu +3\sigma )$.

$(\mu -3\sigma ,\mu +3\sigma )$$=(24.85-18.69,24.85+18.69)$

role="math" localid="1653127938291" $=(6.16,43.54)$

Hence, approximately $99.7\%$of all such children watch between $\underset{¯}{6.16}$and $\underset{¯}{43.54}$hours of TV per week.

Part (d) Step 1: Given Information

The average weekly television watching time is $64.4$ hours, with a standard deviation of$24$ hours.

Part (d) Step 2: Explanation

Construct a graph to portray that approximately $68\%$of all such children watch between $18.62$and $31.08$hours of TV per week.

Part (d) Step 3: Explanation

Construct a graph to portray that approximately $95\%$of all such children watch between $12.39$and $37.31\%$hours of TV per week.

Construct a graph to portray that approximately $99.7\%$of all such children watch between $6.16$and $43.54$hours of TV per week.