Solution manual for statistics for managers using microsoft excel 8th edition

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CHAPTER 7

7.3 (a) For samples of 25 customer receipts for a supermarket for a year, the sampling distribution of sample means is the distribution of means from all possible samples of 25 customer receipts for a supermarket for that year.

(b) For samples of 25 insurance payouts in a particular geographical area in a year, the sampling distribution of sample means is the distribution of means from all possible samples of 25 insurance payouts in that particular geographical area in that year.

(c) For samples of 25 Call Center logs of inbound calls tracking handling time for a credit card company during the year, the sampling distribution of sample means is the distribution of means from all possible samples of 25 Call Center logs of inbound calls tracking handling time for a credit card company during that year.

7.4 (a) Sampling Distribution of the Mean for n = 2 (without replacement)

(a) Mean of All Possible Sample Means: Mean of All Population Elements:

Both means are equal to 6. This property is called unbiasedness.

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7.4 (b) Sampling Distribution of the Mean for n = 3 (without replacement) cont.

(c) The distribution for n = 3 has less variability. The larger sample size has resulted in sample means being closer to μ .

(d) (a) Sampling Distribution of the Mean for n = 2 (with replacement)

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250 Chapter 7: Sampling Distributions

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7.4 (d) (a) Mean of All Possible Mean of All cont. Sample Means: Population Elements:

Both means are equal to 6. This property is called unbiasedness.

(b) Repeat the same process for the sampling distribution of the mean for n = 3 (with replacement). There will be 3 6216 = different samples. 6 X

= This is equal to μ , the population mean.

(c) The distribution for n = 3 has less variability. The larger sample size has resulted in more sample means being close to μ

7.5 (a) Because the population diameter of tennis balls is approximately normally distributed, the sampling distribution of samples of 9 will also be approximately normal with a mean of

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7.6 (a) When n = 4 , the shape of the sampling distribution of X should closely resemble the shape of the distribution of the population from which the sample is selected. Because the mean is larger than the median, the distribution of the sales price of new houses is skewed to the right, and so is the sampling distribution of X although it will be less skewed than the population.

(b) If you select samples of n = 100, the shape of the sampling distribution of the sample mean will be very close to a normal distribution with a mean of $345,800 and a standard deviation of X n σ σ = = $9,000.

(c) X σ = 9000 100 90000 = = n σ

PHStat output:

Probability for X <=

X Value370000

Z Value2.6888889

P(X<=370000)0.9964

P( X < 370,000) = P(Z < 2.6889) = 0.9964

(d) PHStat output:

Probability for a Range

From X Value350000

To X Value365000

Z Value for 3500000.4666667

Z Value for 3650002.1333333

P(X<=350000)0.6796

P(X<=365000)0.9836

P(350000<=X<=365000)0.3039

P(350,000 < X < 365,000) = P(0.4667 < Z < 2.1333) = 0.3039

7.7 (a) X σ = 5 1 25 n σ ==

PHStat output:

Probability for a Range

From X Value27.5

To X Value28.5

Z Value for 27.5 ‐ 0.5

Z Value for 28.50.5

P(X<=27.5)0.3085

P(X<=28.5)0.6915

P(27.5<=X<=28.5)0.3829

P(27.5 < X < 28.5) = P(– 0.50 < Z < 0.50) = 0.3829

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7.7 (b) PHStat output: cont.

Probability for a Range

From X Value27

To X Value29

Z Value for 27 ‐ 1

Z Value for 291

P(X<=27)0.1587

P(X<=29)0.8413

P(27<=X<=29)0.6827

P(27 < X < 29) = P(– 1 < Z < 1) = 0.6827

(c) X σ = 5 0.5 100 n σ ==

PHStat output:

Probability for a Range

From X Value27.5

To X Value28.5

Z Value for 27.5 ‐ 1

Z Value for 28.51

P(X<=27.5)0.1587

P(X<=28.5)0.8413

P(27.5<=X<=28.5)0.6827

P(27.5 < X < 28.5) = P(– 1.00 < Z < 1.00) = 0.6827

(d) With the sample size increasing from n = 25 to n = 100, more sample means will be closer to the distribution mean. The standard error of the sampling distribution of size 100 is much smaller than that of size 25, so the likelihood that the sample mean will fall within ±0.5 minutes of the mean is much higher for samples of size 100 (probability = 0.6827) than for samples of size 25 (probability = 0. 3829).

7.8 (a) P( X > 16) = P(Z > – 1.0) = 0.8413

PHStat output:

Probability for X >

X Value16

Z Value ‐ 1

P(X>16)0.8413

(b) PHStat outpuat:

Find X and Z Given Cum. Pctage.

Cumulative Percentage85.00%

Z Value1.0364

X Value18.0364

P( X < A) = P(Z < 1.0364) = 0.85 X = 17 + 1.0364 (1) = 18.0364 hours

(c) To be able to use the standard normal distribution as an approximation for the area under the curve, we must assume that the population is symmetrically distributed such that the central limit theorem will likely hold for samples of n = 16.

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7.12 (d) (a) Partial PHstat output: cont.

(b)

(c)

P(p > 0.55) = P (Z > 2.40) = 1 – 0.9918 = 0.0082

If the sample size is increased to 400, the probably in (a), (b) and (c) is smaller, larger, and smaller, respectively because the standard error of the sampling distribution of the sample proportion becomes smaller and, hence, the sampling distribution is more concentrated around the true population proportion.

7.13 (a) Partial PHstat output:

a Range

(b)

(c)

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7.13 (d) Partial PHstat output: cont.

(b)

P(X<=0.78)0.8191

Probability for a Range

Z Value for 0.7 ‐ 0.911922

Z Value for 0.780.9119215

P(X<=0.7)0.1809

P(X<=0.78)0.8191

P(0.7<=X<=0.78)0.6382

P(0.70 < p < 0.78) = P (-0.9119 < Z < 0.9119) = 0.6382

(c) Partial PHStat output:

X Value0.76

Z Value0.4559608

P(X>0.76)0.3242

Probability for X > P(p > 0.76) = P (Z > 0.4560) = 0.3242

Z Value for 0.7 ‐ 1.823843

Z Value for 0.781.823843 P(X<=0.7)0.0341 P(X<=0.78)0.9659

(b) Partial PHStat output:

From X Value0.5

To X Value0.6

Z Value for 0.5 ‐ 1.005038

Z Value for 0.61.0050378

P(X<=0.5)0.1574

P(X<=0.6)0.8426

P(0.5<=X<=0.6)0.6851

258 Chapter 7: Sampling Distributions

7.15 (c) Partial PHStat output:

Probability for X >

X Value0.6

Z Value1.0050378

P(X>0.6)0.1574

P(p > 0.6) = P(Z > 1.0053) = 0.1574

(d)

0.57,

(a) PHStat output:

Probability for X <=

X Value0.55

Z Value0

P(X<=0.55)0.5000

P(p < 0.55) = P(Z < 0) = 0.5

(b) Partial PHStat output:

Probability for a Range

From X Value0.5

To X Value0.6

Z Value for 0.5 ‐ 2.010076

Z Value for 0.62.0100756

P(X<=0.5)0.0222

P(X<=0.6)0.9778

P(0.5<=X<=0.6)0.9556

= 0.0249

P(0.52 < p < 0.62) = P(-2.0101< Z < 2.0101) = 0.9556

(c) Partial PHStat output:

Probability for X >

X Value0.6

Z Value2.0100756

P(X>0.6)0.0222

P(p > 0.6) = P(Z > 2.0100) = 0.0222

7.16 (a) PHStat output:

Probability for a Range

From X Value0.14

To X Value0.2

Z Value for 0.14 ‐ 1.129465

Z Value for 0.21.1294649

P(X<=0.14)0.1294

P(X<=0.2)0.8706

P(0.14<=X<=0.2)0.7413

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260 Chapter 7: Sampling Distributions

7.17 (c) Partial PHStat output: cont.

P(X<=0.62)0.3393

P(X<=0.67)0.9512

P(0.62<=X<=0.67)0.6119

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-2.4398) = 0.0073

(c) Increasing the sample size by a factor of 4 decreases the standard error by a factor of 4 . The sampling distribution of the proportion becomes more concentrated around the true proportion of 0.30 and, hence, the probability in (b) becomes smaller than that in (a).

7.19 Because the average of all the possible sample means of size n is equal to the population mean.

7.20 The variation of the sample means becomes smaller as larger sample sizes are taken. This is due to the fact that an extreme observation will have a smaller effect on the mean in a larger sample than in a small sample. Thus, the sample means will tend to be closer to the population mean as the sample size increases.

7.21 As larger sample sizes are taken, the effect of extreme values on the sample mean becomes smaller and smaller. With large enough samples, even though the population is not normally distributed, the sampling distribution of the mean will be approximately normally distributed.

7.22 The population distribution is the distribution of a particular variable of interest, while the sampling distribution represents the distribution of a statistic.

7.23 When the items of interest and the items not of interest are at least 5, the normal distribution can be used to approximate the binomial distribution.

262 Chapter 7: Sampling Distributions

7.27 (c) Partial PHStat output: cont.

Given

7.29 (b) Partial PHStat output: cont.

(c)

(e)

Probability for a Range

20)0.0000 P(X<=‐ 10)0.0055 P(

10)0.0055

(f) Partial PHStat output:

Probability for X > P(X> -5) = P(Z > -1.54) = 0.9382

P(X>‐ 5)0.9382

(g) Since the sample mean of returns of a sample of stocks is distributed closer to the population mean than the return of a single stock, the probabilities in (a) and (b) are higher than those in (d) and (e) while the probability in (c) is lower than that in (f).

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