Q.6.106 - Elementary Statistics

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.

Step-by-Step Solution

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 $\underset{¯}{12.39 and }37.31$ 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.

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 $\sigma =6.23 hours$.

Part (a) Step 2: Explanation

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

$(\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. $(\mu -3\sigma ,\mu +3\sigma )$.

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

$=(6.16,43.54)$

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.