Question

According to a recent poll, 64% of adults who use the Internet
have paid to download music. In a random sample of size 675, adults
who use the Internet, let [^(*p*)] represent the sample
proportion who have paid to download music.

Find the mean of the sampling distribution of [^(*p*)].
*Answer to 2 decimal places.*

Tries 0/5 |

Find the standard deviation of the sampling distribution of
[^(*p*)]. *Answer to 4 decimal places.*

Tries 0/5 |

What does the Central Limit Theorem say about the shape of the
sampling distribution of [^(*p*)]?

The Central Limit Theorem says that the shape of the sampling
distribution of [^(*p*)] is completely determined by the
population parameter *p*.

The Central Limit Theorem says that as the sample size grows large,
the sampling distribution of [^(*p*)] will approach the
population distribution.

The Central Limit Theorem says that as the sample size grows large,
the sampling distribution of [^(*p*)] will become
approximately normal.

The Central Limit Theorem does not say anything about the sampling
distribution of [^(*p*)] because it is not a population
mean.

Tries 0/3 |

Compute the probability that [^(*p*)] is less than 0.598.
*Answer to 4 decimal places.*

Tries 0/5 |

Find *a* such that *P*(0.64 − *a* <
[^(*p*)] < 0.64 +*a* ) = 0.59. *Answer to 3
decimal places.*

Answer #1

p = 0.64, n = 675

Mean of the sampling distribution, μ_{p̂}
= 0.64

Standard deviation of the sampling distribution, σ_{p̂}
= √(p*(1-p)/n) = √(0.64 * 0.36 / 675) = 0.0185

--

The Central Limit Theorem says that as the sample size grows large, the sampling distribution of p̂ will become approximately normal.

--

P(p̂ < 0.598)

= P((p̂ - μ_{p̂})/σ_{p̂} < (0.598 -
0.64)/0.0185)

= P(z < -2.2733)

Using excel function:

= NORM.S.DIST(-2.2733, 1)

= **0.0115**

--

P(0.64-a < p̂ < 0.64+a) = 0.59

= P(p̂ < 0.64+a) - P(p̂ < 0.64-a) = 0.59

= P(p̂ < 0.64+a) - [1 - P(p̂ < 0.64+a)] = 0.59

= 2P(p̂ < 0.64+a) - 1 = 0.59

= P(p̂ < 0.64+a) = 0.795

Z score at p = 0.795 using excel = NORM.S.INV(0.795) = 0.8239

Value of p = µ_{p̂} + z*σ_{p̂} = 0.64 +
(0.8239)*0.0185 = 0.655

**a** = 0.655 - 0.64 = **0.015**

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