Question

Part 1 You have been asked to estimate the mean number of seals killed by a polar bear in one year. The Polar Bear tab of this spreadsheeet contains a random sample of polar bears and their annual seal kills 1. What is your best estimate for the population mean? 2. If you are going to construct at 95% confidence interval, what is the appropriate critical value ("? 13. What is the 95% confidence interval for the population mean? Part 2 You have been asked to confirm or deny the claim that the mean age at which an infant first speaks is greater than 7 months. The Infant Data tab of this spreadsheet contains a random sample of infants that was collected and the age at which they first spoke. You may assume that each infant is independent and that all necessary assumptions are met. Answer the following: 1. What are the null and alterative Hypotheses? 22. What are the sample mean and sample standard deviation? 5 3. What is the value of the test statistict of the sample mean, assuming the nul hypothesis is true? 3 4. What is the P-value for this test? 15. What is your conclusion at a signficance level of 5%?

Answer 1

Part 1:

Based on the given data,

1.

CORREL D M N O P Q Polar Bear 3 Mean - 2 3 Returns the average (arithmetic mean) of its arguments, which can be numbers or names, arrays, or references that contain numbers =avera AVERAGE AVERAGEA AVERAGEIF AVERAGEIFS 5 7 - X fx -avera C Annual Seal Kills 200 266 219 332 203 251 211 268 240 201 244 219 237 209 347 248 321 352 238 268 12 13 16 17 18 19 20

CORREL Polar Bear X fx =AVERAGE(B2:B52 C D F G Annual Seal Kills 200 266 |=AVERAGE(B2:352 219 AVERAGE(number1, [number2], ...) 332 203 251 51R x 10 211 268 240 201 244 219 237 209 347 248 321 352 238 268

The best estimate of population mean ($\mu$) would be the sample mean ($\overline{x}$). Hence, to estimate the mean no. of annual sea kills, we may make use of the given sample of size 51 and compute the  mean no. of annual sea kills in the sample.

We get $\widehat{\mu }=\overline{x}=274.67$

2. The 100 (1-$\alpha$) % CI for population mean $\mu$ can be obtained by the formula:

$\overline{x}\mp t_{\alpha ,n-1}\frac{s}{\sqrt{n}}$

For $\alpha =0.05,n = 51$

letin TINV Returns the inverse of the Student's t-distribution

-TINV(0.05,50 TINV(probability, deg_freedom)

We get the critical value of t as 2.009.

3. Substituting the values:

Mean 274.67 Standard deviation =stdev STDEV Estimates standard deviation based on a sample (ignores logical values and text in the sample) | f: STDEVA STDEVP STDEVPA

Mean Standard deviation 274.67 STDEV(B2:B52 STDEV(number1, (number2], ...)

We get s = 53.71

$274.67\mp 2.009\frac{53.71}{\sqrt{51}}$

= (259.56, 289.78)

Part 2:

1. Let $\mu$ denote the mean age at which the infant first speaks. We have to test:

$H_{0}:\mu \leq 7$ Vs $H_{a}:\mu > 7$

2. From the given data,

$\overline{x}=7.83,s = 2.91,n=81$

3. The appropriate statistical test to test the above hypothesis would be a one sample t test.

But before running this test, we must ensure that the data satisfies the assumptions of this test:

- The data is continuous - The observations are independent - The data is normally distributed - There are no outliers   

Assuming that all the assumptions are satisfied:

The test statistic is given by:

with critical region given by: $t>t_{crit}$ for right tailed test.

a.

The critical value of t is given by:  

Using excel,

Etin TINV Returns the inverse of the Student's t-distribution

(................Since, excel function gives only the two tailed probabilities, we must convert the one tailed 0.05 into two tailed 0.05*2 = 0.10)

ETINV(0.10,80 TINV(probability, deg_freedom)

We get $t_{0.10,80}=1.664$

Substituting the values obtained in the test statistic:

$t=\frac{7.83-7}{2.91/\sqrt{81}}$

= 2.56

4. To compute the p-value:

=tdi TDIST Returns the Student's t-distribution

=TDIST(2.56,80,1 TDIST(x, deg_freedom, tails)

We get p-value = 0.0062

5. Since, the p-value of the test 0.0062 < 0.05, we may reject H0. We may conclude that the data provides sufficient evidence to support the claim that the mean age at which the infant first speaks is greater than 7 months.