Stats HW 5

What are the​ hypotheses? (for 23)

​H0: μ=33 years H1​: μ≠33 years

What are the​ hypotheses?

​H0:μ=210 sec H1​: μ>210 sec

Which of the following statistics are unbiased estimators of population​ parameters?

-Sample variance used to estimate a population variance. -Sample mean used to estimate a population mean. -Sample proportion used to estimate a population proportion.

Identify the​ P-value.

0.001

Identify the​ P-value. (for 23)

0.0214

State the final conclusion that addresses the original claim. Choose the correct answer below. (for 23)

Reject H0. There is sufficient evidence to warrant rejection of the claim that the mean age of actresses when they win an acting award is 33 years.

State the final conclusion that addresses the original claim. Choose the correct answer below.

Reject H0. There is sufficient evidence to support the claim that the sample is from a population of songs with a mean length greater than 210 sec.

Identify the test statistic (for 23)

2.345

Listed below are measured amounts of lead​ (in micrograms per cubic​ meter, or μg/m3​) in the air. The EPA has established an air quality standard for lead of 1.5 μg/m3. The measurements shown below were recorded at a building on different days. Use the given values to construct a 95​% confidence interval estimate of the mean amount of lead in the air. Is there anything about this data set suggesting that the confidence interval might not be very​ good? 5.40 1.10 0.39 0.69 0.71 1.10 A. What is the confidence interval for the population mean μ​? B. Is there anything about this data set suggesting that the confidence interval might not be very​ good?

A) -0.427 μg/m3 < μ < 3.557 (STAT--> T Stats--> One Sample) B) Yes, the value of 5.40 appears to be an outlier.

Assume that​ women's heights are normally distributed with a mean given by μ=63.4 in​, and a standard deviation given by σ=2.7 in. (a) If 1 woman is randomly​ selected, find the probability that her height is less than 64 in. ​(b) If 30 women are randomly​ selected, find the probability that they have a mean height less than 64 in.

A) 0.5879* (Stat-->Calculators--> Normal) Mean: 63.4 Standard deviation: 2.7 P(X ≤ 64)= answer B) 0.8888*

A data set lists earthquake depths. The summary statistics are n=400​, x=4.97 km, s=4.76 km. Use a 0.01 significance level to test the claim of a seismologist that these earthquakes are from a population with a mean equal to 4.00. Assume that a simple random sample has been selected. Identify the null and alternative​ hypotheses, test​ statistic, P-value, and state the final conclusion that addresses the original claim. A)What are the​ hypotheses? B)Identify the test statistic. C)Identify the​ P-value. D)_________ H0 .There is _________ evidence to conclude that the mean of the population of earthquake depths is 4.00 km _______ correct.

A) H0: μ= 4.00 km H1​: μ ≠ 4.00 km B) t= 4.08 C) P-value is 0.000 D) Reject; sufficient; is not

The accompanying data table lists the weights of male college students in kilograms. Test the claim that male college students have a mean weight that is less than the 84 kg mean weight of males in the general population. Use a 0.10 significance level. Identify the null​ hypothesis, alternative​ hypothesis, test​ statistic, P-value, and conclusion for the test. Assume this is a simple random sample. A)What are the​ hypotheses? B)Identify the test statistic. C)Identify the​ P-value. D)State the final conclusion that addresses the original claim. Choose the correct answer below. _________ H0. There is ________ evidence to conclude that male college students have a mean weight that is less than the 84 kg mean weight of males in the general population.

A) H0: μ= 84 kg H1​: μ < 84 kg B) t= -6.128 C) P-value is 0.0001 D) Reject; sufficient

The accompanying data table lists the magnitudes of 50 earthquakes measured on the Richter scale. Test the claim that the population of earthquakes has a mean magnitude greater than 1.00. Use a 0.05 significance level. Identify the null​ hypothesis, alternative​ hypothesis, test​ statistic, P-value, and conclusion for the test. Assume this is a simple random sample. A)What are the​ hypotheses? B)Identify the test statistic. C)Identify the​ P-value. D)State the final conclusion that addresses the original claim. Choose the correct answer below.

A) H0: μ=1.00 in magnitude H1​: μ > 1.00 in magnitude B) t= 2.22 C) P-value is 0.015 D) Reject H0. There is sufficient evidence to conclude that the population of earthquakes has a mean magnitude greater than 1.00.

Assume that a simple random sample has been selected and test the given claim. Identify the null and alternative​ hypotheses, test​ statistic, P-value, and state the final conclusion that addresses the original claim. Listed below are brain volumes in cm3 of unrelated subjects used in a study. Use a 0.01 significance level to test the claim that the population of brain volumes has a mean equal to 1100.3 cm3. 964 1027 1273 1080 1069 1172 1068 1347 1101 1205 A)What are the​ hypotheses? B)Identify the test statistic. C)Identify the​ P-value. D)State the final conclusion that addresses the original claim. Choose the correct answer below.

A) H0: μ=1100.3 cm3 H1​: μ≠1100.3 cm3 B) t= 0.816 C) P-value is 0.4354 D)Fail to reject H0. There is insufficient evidence to warrant rejection of the claim that the population of brain volumes has a mean equal to 1100.3 cm3.

The overhead reach distances of adult females are normally distributed with a mean of 202.5 cm and a standard deviation of 8.9 cm. A. Find the probability that an individual distance is greater than 215.00 cm. b. Find the probability that the mean for 25 randomly selected distances is greater than 200.30 cm. c. Why can the normal distribution be used in part​ (b), even though the sample size does not exceed​ 30?

A. 0.0801* (Stat-->Calculators--> Normal) Mean: 202.5 Standard deviation: 8.9 P(X ≥ 215 )= answer B. 0.8918 (Stat-->Calculators--> Normal) Mean: 202.5 Standard deviation: 1.78 ( 8.9/(25)^1/2) P(X ≥ 215 )= answer C. The normal distribution can be used because the original population has a normal distribution.

The lengths of pregnancies are normally distributed with a mean of 268 days and a standard deviation of 15 days. A.Find the probability of a pregnancy lasting 309 days or longer. B.If the length of pregnancy is in the lowest 4​%, then the baby is premature. Find the length that separates premature babies from those who are not premature.

A. Probability = 0.0031* (Stat-->Calculators--> Normal) Mean: 268 Standard deviation: 15 P(X ≥ 309)= answer B. 242 days* (Stat-->Calculators--> Normal) Mean: 268 Standard deviation: 15 P(X ≤ answer)= 0.04

The population of current statistics students has ages with mean μ and standard deviation σ. Samples of statistics students are randomly selected so that there are exactly 49 students in each sample. For each​ sample, the mean age is computed. What does the central limit theorem tell us about the distribution of those mean​ ages?

Because n>​30, the sampling distribution of the mean ages can be approximated by a normal distribution with mean μ and standard deviation σ/(49)^1/2. (The central limit theory applies whenever n>30. Because n>​30, the sampling distribution of the mean ages can be approximated by a normal distribution with mean μ and standard deviation σ/(n)^1/2).

Do one of the​ following, as appropriate.​ (a) Find the critical value zα/2​, ​(b) find the critical value tα/2​, ​(c) state that neither the normal nor the t distribution applies. Confidence level 90​%; n=17​; σ is known​; population appears to be very skewed.

Neither normal nor t distribution applies

Do one of the​ following, as appropriate.​ (a) Find the critical value zα/2​, ​(b) find the critical value tα/2​, ​(c) state that neither the normal nor the t distribution applies. Confidence level 99​%; n=27​; σ is known​; population appears to be very skewed.

Neither normal nor t distribution applies

Assume that adults have IQ scores that are normally distributed with a mean of 103 and a standard deviation of 15. Find the third quartile Q3​, which is the IQ score separating the top​ 25% from the others.

The 3rd quartile, Q3, is *113.1* (Stat-->Calculators--> Normal) Mean: 103 Standard deviation: 15 P(X ≤ ____)= 0.75

Assume that adults have IQ scores that are normally distributed with a mean of 103 and a standard deviation of 15. Find the third quartile Q3​, which is the IQ score separating the top​ 25% from the others. area = 0.9

The indicated IQ score, x, is *119.2*. (Stat-->Calculators--> Normal) Mean: 100 Standard deviation: 15 P(X ≤ ____)= 0.90

What do the results suggest about the advice given in the​ manual?

The results suggest that the advice of writing a song that must be no longer than 210 seconds is not sound advice.

One of the tallest living men has a height of 242 cm. One of the tallest living women is 227 cm tall. Heights of men have a mean of 179 cm and a standard deviation of 6 cm. Heights of women have a mean of 166 cm and a standard deviation of 5 cm. Relative to the population of the same gender, who is​ taller? Explain.

The woman is relatively taller because the z score for her height is greater than the z score for the man​'s height.

To construct a confidence interval using the given confidence​ level, do whichever of the following is appropriate.​ (a) Find the critical value zα/2​, ​(b) find the critical value tα/2​, or​ (c) state that neither the normal nor the t distribution applies. 90​%; n=400​; σ=16.0​; population appears to be skewed

Zα/2 = 1.645

To construct a confidence interval using the given confidence​ level, do whichever of the following is appropriate.​ (a) Find the critical value zα/2​, ​(b) find the critical value tα/2​, or​ (c) state that neither the normal nor the t distribution applies. 95​%; n=100​; σ=14.0​; population appears to be skewed

Zα/2 = 1.96

A data set includes 103 body temperatures of healthy adult humans for which x=98.3°F and s=0.73°F. a)The best point estimate is ______. B)Using the sample​ statistics, construct a 99​% confidence interval estimate of the mean body temperature of all healthy humans. Do the confidence interval limits contain 98.6°​F? What does the sample suggest about the use of 98.6°F as the mean body​ temperature? What is the confidence interval estimate of the population mean μ​? C)Do the confidence interval limits contain 98.6°​F? D) What does this suggest about the use of 98.6°F as the mean body​ temperature?

a) 98.3°F b) 98.111 F < u < 98.489 F c) No d) This suggests that the mean body temperature could be lower than 98.6°F.

Use technology and the given confidence level and sample data to find the confidence interval for the population mean μ. Assume that the population does not exhibit a normal distribution. Weight lost on diet: 95% confidence, n=61, x=2.0 kg, s=4.3 kg a. What is the confidence interval for the population mean μ​? b. Is the confidence interval affected by the fact that the data appear to be from a population that is not normally​ distributed?

a. 0.9 kg< μ <3.1 kg (STAT--> T Stats--> One Sample) b. No, because the sample size is large enough.

Find the area of the shaded region. The graph to the right depicts IQ scores of​ adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15.

area= *0.8610* (Stat-->Calculators--> Normal) Mean: 100 Standard deviation: 15 P (75 ≤ X ≤ 120)= 0.8610

Identify the test statistic.

t= 3.30