Determine whether each of the following statements is true or false, and explain why in a few sentences.

1. The mean, median, and mode of a normal distribution are all equal.

2. If the mean, median, and mode of a distribution are all equal, then the distribution must be a normal distribution.

3. If the means of two distributions are equal, then the variance must also be equal.

4. The sample mean is not the same as the population mean.

5. The mode of a distribution is the middle element of the distribution.

6. A large variance indicates that the data are grouped closely together.

Answer the following questions in a few sentences.

7. What is meant by the range of a distribution?

8. How are the variance and the standard deviation of a distribution related? What is measured by the standard deviation?

9. Describe the characteristics of a normal distribution.

10. What is meant by a skewed distribution?

Watch Corporation of Switzerland claims that its watches on average will neither gain nor lose time during a week. A sample of 18 watches provided the following gains (+) or losses (−) in seconds per week.

Data File

a)State the null hypothesis and the alternate hypothesis. State the decision rule for 0.02 significance level. (Negative amounts should be indicated by a minus sign. Round your answers to 3 decimal places.)

b)Compute the value of the test statistic. (Negative amount should be indicated by a minus sign. Round your answer to 3 decimal places.)

c)Is it reasonable to conclude that the mean gain or loss in time for the watches is 0? Use the 0.02 significance level.

d)Estimate the p-value.

Problem 4.3.8.

In order to guarantee smooth operation, the University has three web- servers. Each can handle the traffic by itself, and the probability that each is not working on a given day is 10%, independently of the other servers. Assuming that the system is up, what is the probability that only one server is functioning?

Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther.† Suppose a small group of 20 Allen's hummingbirds has been under study in Arizona. The average weight for these birds is x = 3.15 grams. Based on previous studies, we can assume that the weights of Allen's hummingbirds have a normal distribution, with σ = 0.34 gram.

(a) Find an 80% confidence interval for the average weights of Allen's hummingbirds in the study region. What is the margin of error? (Round your answers to two decimal places.)

lower limit:

upper limit:

margin of error:

(b) What conditions are necessary for your calculations? (Select all that apply.)

uniform distribution of weights

σ is unknown

n is large

normal distribution of weights

σ is known

(c) Interpret your results in the context of this problem.

The probability to the true average weight of Allen's hummingbirds is equal to the sample mean.

The probability that this interval contains the true average weight of Allen's hummingbirds is 0.80.

The probability that this interval contains the true average weight of Allen's hummingbirds is 0.20.

There is a 20% chance that the interval is one of the intervals containing the true average weight of Allen's hummingbirds in this region.

There is an 80% chance that the interval is one of the intervals containing the true average weight of Allen's hummingbirds in this region.

(d) Find the sample size necessary for an 80% confidence level with a maximal margin of error E = 0.09 for the mean weights of the hummingbirds. (Round up to the nearest whole number.)

You wish to test the following claim ( H a ) at a significance level of α = 0.001 .

H o : μ = 77.2

H a : μ < 77.2

You believe the population is normally distributed, but you do not know the standard deviation. You obtain a sample of size n = 16 with mean M = 65.4 and a standard deviation of S D = 15.2 .

What is the p-value for this sample? (Report answer accurate to four decimal places.)

p-value =

The p-value is...

less than (or equal to) α or greater than α

This p-value leads to a decision to...

reject the null or accept the null or fail to reject the null

As such, the final conclusion is that...

a . There is sufficient evidence to warrant rejection of the claim that the population mean is less than 77.2.

b . There is not sufficient evidence to warrant rejection of the claim that the population mean is less than 77.2.

c . The sample data support the claim that the population mean is less than 77.2.

d . There is not sufficient sample evidence to support the claim that the population mean is less than 77.2.

Given a standard normal variable, what is the probability Z is greater than 1.25? Round to four decimals and use leading zeros.

A study regarding the relationship between age and the amount of pressure sales personnel feel in relation to their jobs revealed the following sample information. At the 0.02 significance level, is there a relationship between job pressure and age?

State the decision rule. Use 0.02 significance level. (Round your answer to 3 decimal places.)

H₀: Age and pressure are not related.

H₁: Age and pressure are related.

Compute the value of chi-square. (Round your answer to 3 decimal places.)

What is your decision regarding H₀?

The null and alternate hypotheses are:

H₀ : μ_d ≤ 0

H₁ : μ_d > 0

The following sample information shows the number of defective units produced on the day shift and the afternoon shift for a sample of four days last month.

At the 0.100 significance level, can we conclude there are more defects produced on the day shift? Hint: For the calculations, assume the day shift as the first sample.

1. State the decision rule. (Round your answer to 2 decimal places.)

2. Compute the value of the test statistic. (Round your answer to 3 decimal places.)

3. What is the p-value?

• Between 0.025 and 0.05

• Between 0.001 and 0.005

• Between 0.005 and 0.01

4. What is your decision regarding H₀?

• Reject H₀

• Do not reject H₀

• Given the following data where city MPG is the response variable and weight is the explanatory variable, explain why a regression line would be appropriate to analyze the relationship between these variables:

• Construct the regression line for this data.
• Interpret the meaning of the y-intercept and the slope within this scenario.
• What would you predict the city MPG to be for a car that weighs 3000 pounds?
• If a car that weighs 3000 pounds actually gets 32 MPG, would this be unusual? Calculate the residual and talk about what that value represents

NASA is conducting an experiment to find out the fraction of people who black out at G forces greater than 6.

Suppose a sample of 1026 people is drawn. Of these people, 523 passed out. Using the data, construct the 99% confidence interval for the population proportion of people who black out at G forces greater than 6. Round your answers to three decimal places.

Please answer this using Rstudio

For the oyster data, calculate regression fits (simple regression) for the 2D and 3D data

a.1) Give null and alternative hypotheses

a.2) Fit the regression model

a.3) Summarize the fit and evaluation of the regression model (is the linear relationship significant).

a.4 )Calculate residuals and make a qqplot. Is the normal assumption reasonable?

Actual   2D   3D
13.04   47.907   5.136699
11.71   41.458   4.795151
17.42   60.891   6.453115
7.23   29.949   2.895239
10.03   41.616   3.672746
15.59   48.070   5.728880
9.94   34.717   3.987582
7.53   27.230   2.678423
12.73   52.712   5.481545
12.66   41.500   5.016762
10.53   31.216   3.942783
10.84   41.852   4.052638
13.12   44.608   5.334558
8.48   35.343   3.527926
14.24   47.481   5.679636
11.11   40.976   4.013992
15.35   65.361   5.565995
15.44   50.910   6.303198
5.67   22.895   1.928109
8.26   34.804   3.450164
10.95   37.156   4.707532
7.97   29.070   3.019077
7.34   24.590   2.768160
13.21   48.082   4.945743
7.83   32.118   3.138463
11.38   45.112   4.410797
11.22   37.020   4.558251
9.25   39.333   3.449867
13.75   51.351   5.609681
14.37   53.281   5.292105

(5) In a random sample of 21 ​people, the mean commute time to work was 31.5 minutes and the standard deviation was 7.3 minutes. Assume the population is normally distributed and use a​ t-distribution to construct a 80​% confidence interval for the population mean μ. What is the margin of error of μ​? Interpret the results.

(8) Use the standard normal distribution or the​ t-distribution to construct a 99​% confidence interval for the population mean. Justify your decision. If neither distribution can be​ used, explain why. Interpret the results. In a random sample of 17 mortgage​ institutions, the mean interest rate was 3.69​% and the standard deviation was 0.38​%. Assume the interest rates are normally distributed.

(3)

Construct the indicated confidence interval for the population mean μ using the​ t-distribution. Assume the population is normally distributed. c= 0.99​, overbar x=12.9​, s= 4.0​,n=5 round to one decimal place

Using the simple random sample of weights of women from a data​ set, we obtain these sample​ statistics:

nequals=4040

and

x overbarxequals=147.53

lb. Research from other sources suggests that the population of weights of women has a standard deviation given by

sigmaσequals=32.57

lb.

a. Find the best point estimate of the mean weight of all women.

b. Find a 95% confidence interval estimate of the mean weight of all women.

Refer to the accompanying data set of mean​ drive-through service times at dinner in seconds at two fast food restaurants. Construct a 95​% confidence interval estimate of the mean​ drive-through service time for Restaurant X at​ dinner; then do the same for Restaurant Y. Compare the results.

Restaurant X Restaurant Y

80                   101

119                 125

118                 149

151                 117

268                 175

182                 137

120                 111

153                 126

164                 127

213                 123

329                 137

312                 133

182                 226

117                 209

154                 288

143                 128

101                 98

232                 136

240                 247

181                 140

149                 146

195                 202

168                 146

124                 139

69                   136

206                 143

177                 154

114                 138

139                 165

169                 134

193                 240

197                 235

235                 251

190                 237

354                 234

305                 172

206                 86

194                 104

184                 53

192                 172

102                 82

150                 140

174                 145

153                 98

173                 125

159                 146

170                 133

120                 184

135                 152

313                 132

Consider the following data for a dependent variable y and two independent variables, x₁ and x₂.

Round your all answers to two decimal places. Enter negative values as negative numbers, if necessary.

a. Develop an estimated regression equation relating y to x₁.

ŷ =_________ +___________ x₁

Predict y if x₁ = 45.

ŷ = ____________

b. Develop an estimated regression equation relating y to x₂.

ŷ =__________ +____________ x₂

Predict y if x₂ = 15.

ŷ = ___________

c. Develop an estimated regression equation relating y to x₁ and x₂.

ŷ =________ +___________ x1________ +____________ x2

Predict y if x₁ = 45 and x₂ = 15.

ŷ = __________