Question

A market research firm conducts telephone surveys with a 44% historical response rate. What is the probability that in a new sample of 400 telephone numbers, at least 150 individuals will cooperate and respond to the questions?

Answer 1

Answer:

So, the probability is P=0.9953.

Step-by-step explanation:

We know  that a market research firm conducts telephone surveys with a 44% historical response rate.

We get that:

$p=44\%=0.44=\mu_{\hat{x}}\\\\n=400\\\\\hat{p}=\frac{150}{400}=0.375$

We calculate the standar deviation:

$\sigma_{\hat{p}}=\sqrt{\frac{0.44(1-0.44)}{400}}\\\\\sigma_{\hat{p}}=0.025$

So, we get

$z=\frac{\hat{p}-\mu_{\hat{p}}}{\sigma{\hat{p}}}\\\\z=\frac{0.375-0.44}{0.025}}\\\\z=-2.6$

We use a probability table to calculate it

$P(\hat{p}>0.375)=P(z>-2.6)=1-P(z<-2.6)=1-0.0047=0.9953$

So, the probability is P=0.9953.

Answer 2

The probability that in a new sample of 400 telephone numbers, at least 150 individuals will cooperate and respond is 0.9953.

What is probability?

The probability tells us the chances of an event occurring.

As it is given that the market research firm conducts telephone surveys with a 44% historical response rate. Therefore,

$p=44\%=0.44=\mu_x\\\\n=400\\\\\hat{p}=\dfrac{150}{400}=0.375$

The standard deviation is given by the formula,

$\sigma=\sqrt{\dfrac{p(1-p)}{n}}$

Substitute the values we will get,

$\sigma_{\hat{p}}=\sqrt{\dfrac{0.44(1-0.44)}{400}}\\\\\sigma_{\hat{p}}=\sqrt{\dfrac{0.44(0.56)}{400}}\\\\\sigma_{\hat{p}}=0.025$

Now, using the Z-table,

$z=\dfrac{\hat{p}-\mu_{\hat{p}}}{\sigma_{\hat{p}}}$

Substitute the values,

$z=\dfrac{0.375-0.44}{0.025}\\\\z = -2.6\\\\\rm Further,\\\\P(\hat{p} > 0.375)\\\\=P(z > -2.6)\\\\=1-P(z < -2.6)\\\\=1-0.0047\\\\=0.9953\\\\=99.53\%$

Hence,  the probability that in a new sample of 400 telephone numbers, at least 150 individuals will cooperate and respond is 0.9953.

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