Week 7 Homework Assignment

 

1. Determine 

${\mu }_{\overline{x }} and {\sigma }_{\overline{x}}$

  from the given parameters of the population and sample size.

 

$\mu = 83, \sigma = 28, n =49$

 

 

2.  Determine 

${\mu }_{\overline{x }} and {\sigma }_{\overline{x}}$

  and  from the given parameters of the population and sample size.

$\mu = 42, \sigma = 24, n=24$

 

 

3.  A simple random sample of size n = 49 is obtained from a population with mean = 88 and standard deviation = 14 .
​(a) Describe the sampling distribution of

$\overline{x}$

.
​(b) What is 

$P (\overline{x } > 91.6)?$

​?
​(c) What is

$P (\overline{x}\le 83.1)?$

​?
​(d) What is

$P (86.3 < \overline{x} 91.5)?$

​?

​(a) Choose the correct description of the shape of the sampling distribution of

$\overline{x}$

.

The distribution is approximately normal.

The distribution is uniform.

The distribution is skewed right.

The distribution is skewed left.

The shape of the distribution is unknown.

Find the mean and standard deviation of the sampling distribution of .

${\mu }_{\overline{x}} =$

${\sigma }_{\overline{x}} =$

  

 

 

4.    A simple random sample of size n =15  is obtained from a population with mean =63  and standard deviation 17 .
​(a) What must be true regarding the distribution of the population in order to use the normal model to compute probabilities involving the sample​ mean? Assuming that this condition is​ true, describe the sampling distribution of

$\overline{x}$

.
​(b) Assuming the normal model can be​ used, determine ​

$P(\overline{x}<67.3)$

.
​(c) Assuming the normal model can be​ used, determine

$P(\overline{x}\ge 64.9)$

.

​(a) What must be true regarding the distribution of the​ population?

The population must be normally distributed.
 
The population must be normally distributed and the sample size must be large.

The sampling distribution must be assumed to be normal.

Since the sample size is large enough, the population distribution does not need to be normal
 

Assuming the normal model can be​ used, describe the sampling distribution 

$\overline{x}$

 

 

5.  The reading speed of second grade students in a large city is approximately​ normal, with a mean of 92 words per minute​ (wpm) and a standard deviation of 10 wpm. Complete parts​ (a) through​ (f).

​(a) What is the probability a randomly selected student in the city will read more than 96  words per​ minute?

The probability is  ______ .  (Round to four decimal places as​ needed.) 

Interpret this probability. Select the correct choice below and fill in the answer box within your choice.

If 100 different students were chosen from this​ population, we would expect ____ to read exactly 96  words per minute.

If 100 different students were chosen from this​ population, we would expect  ______ to read less than 96 words per minute.

If 100 different students were chosen from this​ population, we would expect _____ to read more than  96 words per minute.

​(b) What is the probability that a random sample of  second grade students from the city results in a mean reading rate of more than  words per​ minute?

The probability is _______ .  ​(Round to four decimal places as​ needed.)

Interpret this probability. Select the correct choice below and fill in the answer box within your choice.

If 100 independent samples of n = 11 students were chosen from this​ population, we would expect  ____ ​sample(s) to have a sample mean reading rate of more than  96 words per minute.

If 100 independent samples of n = 11 students were chosen from this​ population, we would expect _____ sample(s) to have a sample mean reading rate of exactly 96  words per minute.

If 100 independent samples of n =11 students were chosen from this​ population, we would expect _______ ​sample(s) to have a sample mean reading rate of less than  96 words per minute.

 

(c) What is the probability that a random sample of 22 second grade students from the city results in a mean reading rate of more than  96 words per​ minute?

The probability is _______ (Round to four decimal places as​ needed.)

Interpret this probability. Select the correct choice below and fill in the answer box within your choice.

If 100 independent samples of n =22 students were chosen from this​ population, we would expect ____ ​sample(s) to have a sample mean reading rate of less than 96 words per minute.

If 100 independent samples of n = 22 students were chosen from this​ population, we would expect ____ sample(s) to have a sample mean reading rate of more than 96 words per minute.

If 100 independent samples of n = 22 students were chosen from this​ population, we would expect _____ ​sample(s) to have a sample mean reading rate of exactly 96 words per minute.

​(d) What effect does increasing the sample size have on the​ probability? Provide an explanation for this result.

Increasing the sample size decreases the probability because

${\sigma }_{\overline{x}}$

 increases as n increases.

Increasing the sample size increases the probability because 

${\sigma }_{\overline{x}}$

  increases as n increases.

Increasing the sample size decreases the probability because

${\sigma }_{\overline{x}}$

 decreases as n increases.

Increasing the sample size increases the probability because 

${\sigma }_{\overline{x}}$

  decreases as n increases.

​(e) A teacher instituted a new reading program at school. After 10 weeks in the​ program, it was found that the mean reading speed of a random sample of 19 second grade students was 94.3 wpm. What might you conclude based on this​ result? Select the correct choice below and fill in the answer boxes within your choice.
​(Type integers or decimals rounded to four decimal places as​ needed.) 

A mean reading rate of 94.3 wpm is not unusual since the probability of obtaining a result of 94.3  wpm or more is ______.  This means that we would expect a mean reading rate of  94.3 or higher from a population whose mean reading rate is 92  in ______ of every 100 random samples of size n =19 students. The new program is not abundantly more effective than the old program.

A mean reading rate of 94.3  wpm is unusual since the probability of obtaining a result of  94.3 wpm or more is _____.  This means that we would expect a mean reading rate of 94.3 or higher from a population whose mean reading rate is 92  in ______ of every 100 random samples of size n =19 students. The new program is abundantly more effective than the old program.

​(f) There is a​ 5% chance that the mean reading speed of a random sample of  22 second grade students will exceed what​ value?

 

 

 

6.   Suppose a geyser has a mean time between eruptions of 87 minutes . Let the interval of time between the eruptions be normally distributed with standard deviation 24 minutes . Complete parts ​(a) through ​(e) below.

​(a) What is the probability that a randomly selected time interval between eruptions is longer than 97 ​minutes?

The probability that a randomly selected time interval is longer than  minutes is approximately ______ .  ​(Round to four decimal places as​ needed.)

​(b) What is the probability that a random sample of 16  time intervals between eruptions has a mean longer than 97  ​minutes?

The probability that the mean of a random sample of  time intervals is more than  minutes is approximately ______ .  ​(Round to four decimal places as​ needed.)

​(c) What is the probability that a random sample of  30 time intervals between eruptions has a mean longer than 97 ​minutes?

The probability that the mean of a random sample of 30  time intervals is more than 97 minutes is approximately _______.  ​(Round to four decimal places as​ needed.)

​(d) What effect does increasing the sample size have on the​ probability? Provide an explanation for this result. Fill in the blanks below. 

If the population mean is less than 97  ​minutes, then the probability that the sample mean of the time between eruptions is greater than 97 minutes _______ because the variability in the sample mean ________  as the sample size  __________. 

 

(e) What might you conclude if a random sample of  time intervals between eruptions has a mean longer than 97  ​minutes? Select all that apply.

The population mean must be less than 87 ​, since the probability is so low.

The population mean must be more than ​87, since the probability is so low.

The population mean is 87 ​, and this is just a rare sampling.

The population mean is ​87, and this is an example of a typical sampling result.

The population mean may be greater than 87 .

The population mean cannot be ​87, since the probability is so low.

The population mean may be less than 87.